| L(s) = 1 | − 3-s + 8·5-s + 17·7-s − 26·9-s − 70·11-s + 61·13-s − 8·15-s + 83·17-s + 19·19-s − 17·21-s − 115·23-s − 61·25-s + 53·27-s − 279·29-s + 72·31-s + 70·33-s + 136·35-s + 34·37-s − 61·39-s + 108·41-s − 192·43-s − 208·45-s + 392·47-s − 54·49-s − 83·51-s − 131·53-s − 560·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.715·5-s + 0.917·7-s − 0.962·9-s − 1.91·11-s + 1.30·13-s − 0.137·15-s + 1.18·17-s + 0.229·19-s − 0.176·21-s − 1.04·23-s − 0.487·25-s + 0.377·27-s − 1.78·29-s + 0.417·31-s + 0.369·33-s + 0.656·35-s + 0.151·37-s − 0.250·39-s + 0.411·41-s − 0.680·43-s − 0.689·45-s + 1.21·47-s − 0.157·49-s − 0.227·51-s − 0.339·53-s − 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + 70 T + p^{3} T^{2} \) |
| 13 | \( 1 - 61 T + p^{3} T^{2} \) |
| 17 | \( 1 - 83 T + p^{3} T^{2} \) |
| 23 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 279 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 + 192 T + p^{3} T^{2} \) |
| 47 | \( 1 - 392 T + p^{3} T^{2} \) |
| 53 | \( 1 + 131 T + p^{3} T^{2} \) |
| 59 | \( 1 + 609 T + p^{3} T^{2} \) |
| 61 | \( 1 + 338 T + p^{3} T^{2} \) |
| 67 | \( 1 + 461 T + p^{3} T^{2} \) |
| 71 | \( 1 + 750 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1177 T + p^{3} T^{2} \) |
| 79 | \( 1 - 22 T + p^{3} T^{2} \) |
| 83 | \( 1 + 810 T + p^{3} T^{2} \) |
| 89 | \( 1 + 476 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1426 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.877689412732737794514107895669, −7.948930520487697621605378567853, −7.69684493300998657292965082617, −5.94750298207995588932253930595, −5.74828005423672165153507695180, −4.92169640887164943393324572291, −3.55455983813344743745970485696, −2.49516331906696751009521800059, −1.48142883843209680048832769095, 0,
1.48142883843209680048832769095, 2.49516331906696751009521800059, 3.55455983813344743745970485696, 4.92169640887164943393324572291, 5.74828005423672165153507695180, 5.94750298207995588932253930595, 7.69684493300998657292965082617, 7.948930520487697621605378567853, 8.877689412732737794514107895669
