| L(s) = 1 | + 6·3-s + 5·5-s − 32·7-s + 9·9-s + 11·11-s − 48·13-s + 30·15-s − 36·17-s − 44·19-s − 192·21-s + 58·23-s + 25·25-s − 108·27-s − 278·29-s − 112·31-s + 66·33-s − 160·35-s + 194·37-s − 288·39-s − 314·41-s + 396·43-s + 45·45-s − 410·47-s + 681·49-s − 216·51-s + 170·53-s + 55·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.72·7-s + 1/3·9-s + 0.301·11-s − 1.02·13-s + 0.516·15-s − 0.513·17-s − 0.531·19-s − 1.99·21-s + 0.525·23-s + 1/5·25-s − 0.769·27-s − 1.78·29-s − 0.648·31-s + 0.348·33-s − 0.772·35-s + 0.861·37-s − 1.18·39-s − 1.19·41-s + 1.40·43-s + 0.149·45-s − 1.27·47-s + 1.98·49-s − 0.593·51-s + 0.440·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 48 T + p^{3} T^{2} \) |
| 17 | \( 1 + 36 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 278 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 194 T + p^{3} T^{2} \) |
| 41 | \( 1 + 314 T + p^{3} T^{2} \) |
| 43 | \( 1 - 396 T + p^{3} T^{2} \) |
| 47 | \( 1 + 410 T + p^{3} T^{2} \) |
| 53 | \( 1 - 170 T + p^{3} T^{2} \) |
| 59 | \( 1 - 404 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 26 T + p^{3} T^{2} \) |
| 71 | \( 1 + 468 T + p^{3} T^{2} \) |
| 73 | \( 1 + 164 T + p^{3} T^{2} \) |
| 79 | \( 1 + 664 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 97 | \( 1 + 1498 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.799909917038339151004597296163, −9.437647828491945592144894283988, −8.696312959684535300605504999081, −7.43279903384645473749509557646, −6.64635304680106807972732046984, −5.57966130633579740618944521610, −3.99040019340408901897945108580, −3.03580417124465095759367105933, −2.15408062391606461499982734850, 0,
2.15408062391606461499982734850, 3.03580417124465095759367105933, 3.99040019340408901897945108580, 5.57966130633579740618944521610, 6.64635304680106807972732046984, 7.43279903384645473749509557646, 8.696312959684535300605504999081, 9.437647828491945592144894283988, 9.799909917038339151004597296163
