| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 35·11-s + 12·12-s − 54·13-s + 14·14-s + 16·16-s − 32·17-s + 18·18-s − 126·19-s + 21·21-s − 70·22-s − 135·23-s + 24·24-s − 108·26-s + 27·27-s + 28·28-s + 21·29-s − 94·31-s + 32·32-s − 105·33-s − 64·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.959·11-s + 0.288·12-s − 1.15·13-s + 0.267·14-s + 1/4·16-s − 0.456·17-s + 0.235·18-s − 1.52·19-s + 0.218·21-s − 0.678·22-s − 1.22·23-s + 0.204·24-s − 0.814·26-s + 0.192·27-s + 0.188·28-s + 0.134·29-s − 0.544·31-s + 0.176·32-s − 0.553·33-s − 0.322·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 35 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 32 T + p^{3} T^{2} \) |
| 19 | \( 1 + 126 T + p^{3} T^{2} \) |
| 23 | \( 1 + 135 T + p^{3} T^{2} \) |
| 29 | \( 1 - 21 T + p^{3} T^{2} \) |
| 31 | \( 1 + 94 T + p^{3} T^{2} \) |
| 37 | \( 1 + 341 T + p^{3} T^{2} \) |
| 41 | \( 1 - 56 T + p^{3} T^{2} \) |
| 43 | \( 1 - 419 T + p^{3} T^{2} \) |
| 47 | \( 1 + 194 T + p^{3} T^{2} \) |
| 53 | \( 1 - 38 T + p^{3} T^{2} \) |
| 59 | \( 1 - 382 T + p^{3} T^{2} \) |
| 61 | \( 1 + 128 T + p^{3} T^{2} \) |
| 67 | \( 1 - 801 T + p^{3} T^{2} \) |
| 71 | \( 1 + 415 T + p^{3} T^{2} \) |
| 73 | \( 1 + 608 T + p^{3} T^{2} \) |
| 79 | \( 1 - 511 T + p^{3} T^{2} \) |
| 83 | \( 1 + 374 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1234 T + p^{3} T^{2} \) |
| 97 | \( 1 + 182 T + p^{3} T^{2} \) |
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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
−9.028004503609270050891531254464, −8.152681702431770339195324430864, −7.48352524399920711188124812865, −6.58789211333199102580155616627, −5.50452965214854875784142078253, −4.66069840897448904844923691471, −3.85403920065451913997855426642, −2.55726049030507094437001254697, −1.98516758300018896733301570186, 0,
1.98516758300018896733301570186, 2.55726049030507094437001254697, 3.85403920065451913997855426642, 4.66069840897448904844923691471, 5.50452965214854875784142078253, 6.58789211333199102580155616627, 7.48352524399920711188124812865, 8.152681702431770339195324430864, 9.028004503609270050891531254464
