| L(s) = 1 | − 16·13-s − 44·25-s − 112·37-s + 84·49-s − 304·61-s − 104·73-s + 72·97-s − 592·109-s + 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 516·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.23·13-s − 1.75·25-s − 3.02·37-s + 12/7·49-s − 4.98·61-s − 1.42·73-s + 0.742·97-s − 5.43·109-s + 3.47·121-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 − 3.05·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1176508403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1176508403\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 p T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 174 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 546 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1654 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1418 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2802 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3350 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3406 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5394 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 3702 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 14050 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.18102686710989030614557826799, −6.15467368972563569625048305076, −5.82908937713698727993330295431, −5.47461825129319059647224827109, −5.40779196146490128799794546042, −5.14450316316397470035323669332, −5.06822274678705904002383288159, −4.56352036579030033409239311682, −4.56188431075377734340922051989, −4.33943079033002599979898731955, −4.25045981991544227666341731161, −3.57795477640566694159895759024, −3.56486009815332256011269847509, −3.48952358274270554625720193615, −3.33114278151335784027219914625, −2.72274782627156850661667108940, −2.55798766956239175810603131567, −2.34058607112579379910572704391, −2.31952847104946078522629684214, −1.68434288894091451819616701604, −1.49855655119243905425081630151, −1.38689434686113213569854057134, −1.03657755234955879368478042388, −0.22558777061047336091800918224, −0.094727468101646767921113531425,
0.094727468101646767921113531425, 0.22558777061047336091800918224, 1.03657755234955879368478042388, 1.38689434686113213569854057134, 1.49855655119243905425081630151, 1.68434288894091451819616701604, 2.31952847104946078522629684214, 2.34058607112579379910572704391, 2.55798766956239175810603131567, 2.72274782627156850661667108940, 3.33114278151335784027219914625, 3.48952358274270554625720193615, 3.56486009815332256011269847509, 3.57795477640566694159895759024, 4.25045981991544227666341731161, 4.33943079033002599979898731955, 4.56188431075377734340922051989, 4.56352036579030033409239311682, 5.06822274678705904002383288159, 5.14450316316397470035323669332, 5.40779196146490128799794546042, 5.47461825129319059647224827109, 5.82908937713698727993330295431, 6.15467368972563569625048305076, 6.18102686710989030614557826799
