| L(s) = 1 | + 12·3-s + 148·7-s + 63·9-s − 32·13-s + 748·19-s + 1.77e3·21-s − 125·25-s − 216·27-s − 2.85e3·31-s + 544·37-s − 384·39-s − 3.87e3·43-s + 1.16e4·49-s + 8.97e3·57-s + 4.46e3·61-s + 9.32e3·63-s − 1.08e4·67-s − 1.07e3·73-s − 1.50e3·75-s + 1.99e4·79-s − 7.69e3·81-s − 4.73e3·91-s − 3.42e4·93-s − 2.14e4·97-s − 1.84e4·103-s − 1.10e4·109-s + 6.52e3·111-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 3.02·7-s + 7/9·9-s − 0.189·13-s + 2.07·19-s + 4.02·21-s − 1/5·25-s − 0.296·27-s − 2.96·31-s + 0.397·37-s − 0.252·39-s − 2.09·43-s + 4.84·49-s + 2.76·57-s + 1.20·61-s + 2.34·63-s − 2.41·67-s − 0.201·73-s − 0.266·75-s + 3.19·79-s − 1.17·81-s − 0.571·91-s − 3.95·93-s − 2.27·97-s − 1.74·103-s − 0.926·109-s + 0.529·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.628774220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.628774220\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - 4 p T + p^{4} T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 74 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 14702 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 16 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 136622 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 374 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 127502 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 495058 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 46 p T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 272 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5046002 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1936 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 522418 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14329618 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1036142 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2234 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5416 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 48401282 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 538 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9962 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 73243022 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97824962 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10726 T + p^{4} T^{2} )^{2} \) |
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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
−14.66221395333871068111487361959, −14.22887717715379595461614272177, −13.59892741756261102692471502699, −13.36813395193165950254041166615, −12.15033795613636892635739044777, −11.81830047476565230931379464434, −11.05799598841465465000306636463, −10.92109647453883590139901248163, −9.750985767255343320443951933949, −9.253162688969037352368729450046, −8.528811320984105369040825635521, −8.074188389172065274444975566475, −7.59959842648319532680690991183, −7.17089722504430394787668415628, −5.40192465315547569642039337665, −5.20836742358594184165773217420, −4.20346599190881746508898464370, −3.26790150634760412931986254023, −1.98891227612222272111811970475, −1.46503585619444844852222250289,
1.46503585619444844852222250289, 1.98891227612222272111811970475, 3.26790150634760412931986254023, 4.20346599190881746508898464370, 5.20836742358594184165773217420, 5.40192465315547569642039337665, 7.17089722504430394787668415628, 7.59959842648319532680690991183, 8.074188389172065274444975566475, 8.528811320984105369040825635521, 9.253162688969037352368729450046, 9.750985767255343320443951933949, 10.92109647453883590139901248163, 11.05799598841465465000306636463, 11.81830047476565230931379464434, 12.15033795613636892635739044777, 13.36813395193165950254041166615, 13.59892741756261102692471502699, 14.22887717715379595461614272177, 14.66221395333871068111487361959
