| L(s) = 1 | − 2·9-s + 12·17-s + 12·41-s − 28·49-s − 4·73-s + 9·81-s − 36·89-s + 40·97-s − 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.91·17-s + 1.87·41-s − 4·49-s − 0.468·73-s + 81-s − 3.81·89-s + 4.06·97-s − 3.38·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610357683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610357683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−6.67634766185253017076038313156, −6.49658528291289484565504829893, −6.19968026574340729958235694751, −6.02812811742312240303260625299, −5.73500833370211065876594668818, −5.72124185109365549414477044976, −5.46933108977515997538422189439, −5.18327542363076171924691428659, −5.10704055878021143062327891355, −4.64275188382580298493315496524, −4.57464141933300830044398006148, −4.25126532216924503449345604513, −4.13279344236918938774005565884, −3.59542219687515530553426964008, −3.40249798559737535107142791927, −3.27683334603512466694953142117, −3.19480130576218244954632182829, −2.70193516356362930890614125162, −2.67765571868344189571972371844, −2.06475766542034979776729449930, −1.94167666887771177989016513276, −1.47091777749652975528380604914, −1.07943086614518558522052759437, −0.998474776735294409575130738536, −0.24896566273587884968868717262,
0.24896566273587884968868717262, 0.998474776735294409575130738536, 1.07943086614518558522052759437, 1.47091777749652975528380604914, 1.94167666887771177989016513276, 2.06475766542034979776729449930, 2.67765571868344189571972371844, 2.70193516356362930890614125162, 3.19480130576218244954632182829, 3.27683334603512466694953142117, 3.40249798559737535107142791927, 3.59542219687515530553426964008, 4.13279344236918938774005565884, 4.25126532216924503449345604513, 4.57464141933300830044398006148, 4.64275188382580298493315496524, 5.10704055878021143062327891355, 5.18327542363076171924691428659, 5.46933108977515997538422189439, 5.72124185109365549414477044976, 5.73500833370211065876594668818, 6.02812811742312240303260625299, 6.19968026574340729958235694751, 6.49658528291289484565504829893, 6.67634766185253017076038313156
