| L(s) = 1 | − 6·9-s + 32·29-s + 200·41-s + 172·49-s − 104·61-s + 27·81-s + 344·89-s − 128·101-s − 520·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 508·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 1.10·29-s + 4.87·41-s + 3.51·49-s − 1.70·61-s + 1/3·81-s + 3.86·89-s − 1.26·101-s − 4.77·109-s − 0.165·121-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 − 3.00·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.580304745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.580304745\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 254 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 494 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 286 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 638 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2066 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4862 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6710 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )^{2}( 1 + 122 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 5806 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 + 94 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 3550 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 86 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 6722 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.74316956613087047397672367373, −6.51741139084481164557891572995, −6.17765503927852061894092313660, −6.13371412312643394948694235575, −5.94433430083636227326893424492, −5.65456438914869956351330913368, −5.44130453093342136289599551871, −5.34068819485486013700415451222, −4.97084337136529530208760746942, −4.58030929937929010275332014538, −4.46724470019835044366341367451, −4.35437692287545159859252850926, −3.90439512460461375445620699840, −3.74425715717684652509867698452, −3.68621225921823126169892495292, −3.07930645121568068364948672880, −2.71223115639037602565278185703, −2.65059256288880120732877730638, −2.60701123354635050177998409440, −2.13417240418404434657502447382, −1.82402197824797131408684102663, −1.18554592164764739387880470825, −1.09343677286255669572763725313, −0.57852596920097989732799426221, −0.44929959724500706533924568088,
0.44929959724500706533924568088, 0.57852596920097989732799426221, 1.09343677286255669572763725313, 1.18554592164764739387880470825, 1.82402197824797131408684102663, 2.13417240418404434657502447382, 2.60701123354635050177998409440, 2.65059256288880120732877730638, 2.71223115639037602565278185703, 3.07930645121568068364948672880, 3.68621225921823126169892495292, 3.74425715717684652509867698452, 3.90439512460461375445620699840, 4.35437692287545159859252850926, 4.46724470019835044366341367451, 4.58030929937929010275332014538, 4.97084337136529530208760746942, 5.34068819485486013700415451222, 5.44130453093342136289599551871, 5.65456438914869956351330913368, 5.94433430083636227326893424492, 6.13371412312643394948694235575, 6.17765503927852061894092313660, 6.51741139084481164557891572995, 6.74316956613087047397672367373
