| L(s) = 1 | − 9·9-s + 124·11-s − 64·16-s − 200·19-s + 20·29-s − 96·31-s − 496·41-s − 49·49-s − 240·59-s + 1.24e3·61-s − 1.35e3·71-s − 1.48e3·79-s + 81·81-s − 400·89-s − 1.11e3·99-s + 464·101-s + 180·109-s + 8.87e3·121-s + 127-s + 131-s + 137-s + 139-s + 576·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.39·11-s − 16-s − 2.41·19-s + 0.128·29-s − 0.556·31-s − 1.88·41-s − 1/7·49-s − 0.529·59-s + 2.61·61-s − 2.26·71-s − 2.10·79-s + 1/9·81-s − 0.476·89-s − 1.13·99-s + 0.457·101-s + 0.158·109-s + 6.66·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1/3·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.995491165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995491165\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 550 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2770 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22570 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 40790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 248 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 154390 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 102670 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 231190 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 622 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 215690 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 678 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 365870 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 740 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 924550 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 200 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 222590 T^{2} + p^{6} T^{4} \) |
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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
−11.06599068557064614644669511188, −10.01636620284780058560548823370, −9.891277397216770320676781197489, −9.136627175657946579941832332024, −8.916941971575279439224875577701, −8.508379259442428141194372983117, −8.380191019279002068946181413832, −7.16649272725830390843696050666, −7.02725612334736470774967569769, −6.53676720341667882133822843637, −6.21884620100085615494131017259, −5.78781640174166535767489222834, −4.82227763711005961380477982938, −4.38873709311673431788596034395, −3.92720484187089185459946482951, −3.60999989998045946874408501070, −2.66305426886680490382078971294, −1.78704493933456862957921465581, −1.53388401259014487712243343486, −0.43175895752652985751343205627,
0.43175895752652985751343205627, 1.53388401259014487712243343486, 1.78704493933456862957921465581, 2.66305426886680490382078971294, 3.60999989998045946874408501070, 3.92720484187089185459946482951, 4.38873709311673431788596034395, 4.82227763711005961380477982938, 5.78781640174166535767489222834, 6.21884620100085615494131017259, 6.53676720341667882133822843637, 7.02725612334736470774967569769, 7.16649272725830390843696050666, 8.380191019279002068946181413832, 8.508379259442428141194372983117, 8.916941971575279439224875577701, 9.136627175657946579941832332024, 9.891277397216770320676781197489, 10.01636620284780058560548823370, 11.06599068557064614644669511188
