| L(s) = 1 | + 2·2-s + 2·4-s + 6·5-s − 8·7-s + 12·10-s + 10·11-s − 16·14-s − 4·16-s − 8·17-s + 12·20-s + 20·22-s + 22·23-s + 11·25-s − 16·28-s + 26·31-s − 8·32-s − 16·34-s − 48·35-s − 54·37-s + 46·41-s + 114·43-s + 20·44-s + 44·46-s − 78·47-s + 32·49-s + 22·50-s − 136·53-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s + 6/5·5-s − 8/7·7-s + 6/5·10-s + 0.909·11-s − 8/7·14-s − 1/4·16-s − 0.470·17-s + 3/5·20-s + 0.909·22-s + 0.956·23-s + 0.439·25-s − 4/7·28-s + 0.838·31-s − 1/4·32-s − 0.470·34-s − 1.37·35-s − 1.45·37-s + 1.12·41-s + 2.65·43-s + 5/11·44-s + 0.956·46-s − 1.65·47-s + 0.653·49-s + 0.439·50-s − 2.56·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.102956732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.102956732\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 697 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1057 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 114 T + 6498 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 136 T + 9248 T^{2} + 136 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 263 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 58 T + 1682 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 59 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56 T + 1568 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12158 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 174 T + 15138 T^{2} + 174 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6959 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 102 T + 5202 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.17615671874295999566296430522, −9.758096378255554660904835485857, −9.538580368812461235574535920651, −9.072371723758230826241173397655, −8.749774650424794971543528409464, −8.236494702517588838089344087209, −7.37959447138696069598871083219, −7.12874174093308699229249565771, −6.42132027641231522454548101615, −6.41054047614102960252961003009, −5.88907015735098456235621798310, −5.50989553774056507666835365149, −4.76004650684263802849646299575, −4.57936599992913891654289873649, −3.69993933398751357400911758216, −3.45952292515567660004666934878, −2.73241658336069607946109468307, −2.29790457327541422453340210015, −1.52036042890359940443992272401, −0.64949179297125838981370753080,
0.64949179297125838981370753080, 1.52036042890359940443992272401, 2.29790457327541422453340210015, 2.73241658336069607946109468307, 3.45952292515567660004666934878, 3.69993933398751357400911758216, 4.57936599992913891654289873649, 4.76004650684263802849646299575, 5.50989553774056507666835365149, 5.88907015735098456235621798310, 6.41054047614102960252961003009, 6.42132027641231522454548101615, 7.12874174093308699229249565771, 7.37959447138696069598871083219, 8.236494702517588838089344087209, 8.749774650424794971543528409464, 9.072371723758230826241173397655, 9.538580368812461235574535920651, 9.758096378255554660904835485857, 10.17615671874295999566296430522
