| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s − 2·9-s + 10-s + 3·11-s − 12-s − 2·14-s + 15-s + 16-s + 8·17-s + 2·18-s − 20-s − 2·21-s − 3·22-s + 4·23-s + 24-s − 4·25-s + 5·27-s + 2·28-s − 29-s − 30-s + 3·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.94·17-s + 0.471·18-s − 0.223·20-s − 0.436·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.962·27-s + 0.377·28-s − 0.185·29-s − 0.182·30-s + 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.45790780600423535487280017873, −6.64381446645655038387582814769, −6.12588233717155909154310932814, −5.20740025696206682942051926852, −4.85398952064887263701306944035, −3.53368230693309007128803898698, −3.19218601281688441638785666758, −1.78887385760705080248268112882, −1.14414427336107706043571430115, 0,
1.14414427336107706043571430115, 1.78887385760705080248268112882, 3.19218601281688441638785666758, 3.53368230693309007128803898698, 4.85398952064887263701306944035, 5.20740025696206682942051926852, 6.12588233717155909154310932814, 6.64381446645655038387582814769, 7.45790780600423535487280017873
