| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 5·11-s − 12-s − 13-s + 16-s + 2·17-s − 18-s − 7·19-s + 5·22-s + 3·23-s + 24-s + 26-s − 27-s + 6·31-s − 32-s + 5·33-s − 2·34-s + 36-s − 5·37-s + 7·38-s + 39-s + 9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.60·19-s + 1.06·22-s + 0.625·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 1.07·31-s − 0.176·32-s + 0.870·33-s − 0.342·34-s + 1/6·36-s − 0.821·37-s + 1.13·38-s + 0.160·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.52325773225168810034392978554, −7.06198197299263221857528557264, −6.10764713142644633029389131888, −5.63986404565840430884362594448, −4.79392731675998276737044867834, −4.05862912392876506620614432191, −2.79011438120598530431785657624, −2.28980077772198110680333863657, −1.00519392896843933407282265081, 0,
1.00519392896843933407282265081, 2.28980077772198110680333863657, 2.79011438120598530431785657624, 4.05862912392876506620614432191, 4.79392731675998276737044867834, 5.63986404565840430884362594448, 6.10764713142644633029389131888, 7.06198197299263221857528557264, 7.52325773225168810034392978554
