| L(s) = 1 | − 4-s − 6·9-s + 16-s + 8·17-s + 8·23-s − 25-s − 16·29-s + 6·36-s − 12·43-s + 5·49-s − 18·53-s − 28·61-s − 64-s − 8·68-s + 20·79-s + 27·81-s − 8·92-s + 100-s + 2·103-s − 36·107-s − 16·113-s + 16·116-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2·9-s + 1/4·16-s + 1.94·17-s + 1.66·23-s − 1/5·25-s − 2.97·29-s + 36-s − 1.82·43-s + 5/7·49-s − 2.47·53-s − 3.58·61-s − 1/8·64-s − 0.970·68-s + 2.25·79-s + 3·81-s − 0.834·92-s + 1/10·100-s + 0.197·103-s − 3.48·107-s − 1.50·113-s + 1.48·116-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7129323680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7129323680\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 177 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.510048673850909021617120697475, −9.094911053156208925787671110171, −8.974472149526073955237103187319, −8.270018430517927848878049753117, −7.961908680755844214870090117511, −7.66805989713918990186968298301, −7.41553694019543771217817237170, −6.47484245721882194502602386002, −6.43754165048149942360579306641, −5.72066477715818156427266840820, −5.32998179087867168051080862613, −5.32716010435631143558311552002, −4.75028145790022338725931182794, −3.98759213212126458419970751422, −3.47909647332721996099127149099, −3.00934627706136889811015877011, −3.00216269545351557044257099271, −1.89730072661833625261073369982, −1.38069434395769789553772910976, −0.32955312861302782821520521777,
0.32955312861302782821520521777, 1.38069434395769789553772910976, 1.89730072661833625261073369982, 3.00216269545351557044257099271, 3.00934627706136889811015877011, 3.47909647332721996099127149099, 3.98759213212126458419970751422, 4.75028145790022338725931182794, 5.32716010435631143558311552002, 5.32998179087867168051080862613, 5.72066477715818156427266840820, 6.43754165048149942360579306641, 6.47484245721882194502602386002, 7.41553694019543771217817237170, 7.66805989713918990186968298301, 7.961908680755844214870090117511, 8.270018430517927848878049753117, 8.974472149526073955237103187319, 9.094911053156208925787671110171, 9.510048673850909021617120697475
