| L(s) = 1 | − 2·4-s + 8·7-s − 16·13-s + 4·16-s − 32·19-s − 16·28-s + 88·31-s + 68·37-s + 80·43-s − 50·49-s + 32·52-s + 100·61-s − 8·64-s − 16·67-s + 32·73-s + 64·76-s − 152·79-s − 128·91-s − 352·97-s + 56·103-s + 32·112-s − 46·121-s − 176·124-s + 127-s + 131-s − 256·133-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 8/7·7-s − 1.23·13-s + 1/4·16-s − 1.68·19-s − 4/7·28-s + 2.83·31-s + 1.83·37-s + 1.86·43-s − 1.02·49-s + 8/13·52-s + 1.63·61-s − 1/8·64-s − 0.238·67-s + 0.438·73-s + 0.842·76-s − 1.92·79-s − 1.40·91-s − 3.62·97-s + 0.543·103-s + 2/7·112-s − 0.380·121-s − 1.41·124-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.875551883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875551883\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1664 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4160 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15680 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 176 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98629889995339908300020846534, −10.80594052948724572388445384316, −10.06068215057095015217953094954, −9.790436656198777203937546198644, −9.460169139486811842471359441676, −8.660324312559166151955503080685, −8.260640071930376535861670118409, −8.180675030583159019148595418883, −7.47979486438902504691639005918, −7.06028153938759640360044116695, −6.23586915254361910640044429833, −6.10327590941076757315290499347, −5.15798651477903772444393608744, −4.83899316457483005950746687490, −4.23323796104088499016819604541, −4.11438724139480995497635965217, −2.63725051020435494464446810732, −2.62028100789085471413374289907, −1.52612558760377130127707855492, −0.59651829725488660624628487980,
0.59651829725488660624628487980, 1.52612558760377130127707855492, 2.62028100789085471413374289907, 2.63725051020435494464446810732, 4.11438724139480995497635965217, 4.23323796104088499016819604541, 4.83899316457483005950746687490, 5.15798651477903772444393608744, 6.10327590941076757315290499347, 6.23586915254361910640044429833, 7.06028153938759640360044116695, 7.47979486438902504691639005918, 8.180675030583159019148595418883, 8.260640071930376535861670118409, 8.660324312559166151955503080685, 9.460169139486811842471359441676, 9.790436656198777203937546198644, 10.06068215057095015217953094954, 10.80594052948724572388445384316, 10.98629889995339908300020846534
