| L(s) = 1 | − 8·9-s − 176·25-s − 432·49-s − 288·73-s + 132·81-s + 224·97-s − 656·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.04e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 1.40e3·225-s + ⋯ |
| L(s) = 1 | − 8/9·9-s − 7.03·25-s − 8.81·49-s − 3.94·73-s + 1.62·81-s + 2.30·97-s − 5.42·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.15·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 6.25·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2778945361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2778945361\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + 4 T^{2} - 14 p T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| good | 5 | \( ( 1 + 44 T^{2} + 1526 T^{4} + 44 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 7 | \( ( 1 + 108 T^{2} + 7510 T^{4} + 108 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 11 | \( ( 1 + 164 T^{2} + 10838 T^{4} + 164 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 13 | \( ( 1 - 10 p T^{2} + p^{4} T^{4} )^{8} \) |
| 17 | \( ( 1 - 804 T^{2} + 298694 T^{4} - 804 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 19 | \( ( 1 - 444 T^{2} + 199894 T^{4} - 444 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 23 | \( ( 1 - 68 T^{2} + 507590 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 29 | \( ( 1 + 2220 T^{2} + 2330294 T^{4} + 2220 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 276 T^{2} + 817558 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 37 | \( ( 1 - 4548 T^{2} + 8706406 T^{4} - 4548 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 41 | \( ( 1 - 2372 T^{2} + 3650246 T^{4} - 2372 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 - 5052 T^{2} + 12758806 T^{4} - 5052 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 47 | \( ( 1 + 2428 T^{2} + 7825286 T^{4} + 2428 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 53 | \( ( 1 + 7404 T^{2} + 27911414 T^{4} + 7404 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 59 | \( ( 1 + 11332 T^{2} + 54973590 T^{4} + 11332 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 61 | \( ( 1 - 6532 T^{2} + 21106086 T^{4} - 6532 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 - 17596 T^{2} + 117690198 T^{4} - 17596 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 71 | \( ( 1 - 7364 T^{2} + 55381574 T^{4} - 7364 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 73 | \( ( 1 + 36 T + 3494 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{8} \) |
| 79 | \( ( 1 + 9516 T^{2} + 42331798 T^{4} + 9516 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 19364 T^{2} + 181993238 T^{4} + 19364 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 89 | \( ( 1 - 11492 T^{2} + 56580998 T^{4} - 11492 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 97 | \( ( 1 - 28 T + 13814 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.98634087868091521076189233243, −2.81586264768879660838656765911, −2.61381084829120786843213132376, −2.61354046540234182095534932660, −2.46310221390575650289720942241, −2.37956132042494719769879926704, −2.28850714727856581605985229117, −2.21810207132746463110638299644, −1.98121474526673868422548457594, −1.94644430546880114802904520242, −1.92972045022045562489897659419, −1.78672499840155379343505123632, −1.73402319797150705958558248303, −1.61402519250403943970940273920, −1.52167160877160378063390896118, −1.48278573480424621464723944615, −1.36500573372425292324519070883, −1.18524683722328429488665646965, −1.15413403952308972479848764188, −0.61829304095122317236775229696, −0.57111821526368435928969081483, −0.50415426805399766950371831016, −0.32329091844723310516791895937, −0.22752104779488382170965456721, −0.06037910283505228001836728839,
0.06037910283505228001836728839, 0.22752104779488382170965456721, 0.32329091844723310516791895937, 0.50415426805399766950371831016, 0.57111821526368435928969081483, 0.61829304095122317236775229696, 1.15413403952308972479848764188, 1.18524683722328429488665646965, 1.36500573372425292324519070883, 1.48278573480424621464723944615, 1.52167160877160378063390896118, 1.61402519250403943970940273920, 1.73402319797150705958558248303, 1.78672499840155379343505123632, 1.92972045022045562489897659419, 1.94644430546880114802904520242, 1.98121474526673868422548457594, 2.21810207132746463110638299644, 2.28850714727856581605985229117, 2.37956132042494719769879926704, 2.46310221390575650289720942241, 2.61354046540234182095534932660, 2.61381084829120786843213132376, 2.81586264768879660838656765911, 2.98634087868091521076189233243
Plot not available for L-functions of degree greater than 10.
