L(s) = 1 | + 3-s − 2·7-s + 9-s + 4.47·11-s − 4.47·13-s − 4.47·17-s − 2·21-s − 4·23-s + 27-s − 4·29-s + 8.94·31-s + 4.47·33-s + 4.47·37-s − 4.47·39-s + 10·41-s − 4·43-s − 8·47-s − 3·49-s − 4.47·51-s + 4.47·53-s − 13.4·59-s − 10·61-s − 2·63-s − 8·67-s − 4·69-s − 8.94·71-s − 8.94·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 0.333·9-s + 1.34·11-s − 1.24·13-s − 1.08·17-s − 0.436·21-s − 0.834·23-s + 0.192·27-s − 0.742·29-s + 1.60·31-s + 0.778·33-s + 0.735·37-s − 0.716·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 0.428·49-s − 0.626·51-s + 0.614·53-s − 1.74·59-s − 1.28·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s − 1.06·71-s − 1.04·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.85829872364055419138274199540, −7.24687800066375950313695672025, −6.42922593095694567356099724039, −6.03642745594146154833359141072, −4.61632169200811206064256954699, −4.27061044370522814870291213619, −3.22887150114067356712133265076, −2.51012301705214334157245186176, −1.50912943606507894158740600361, 0,
1.50912943606507894158740600361, 2.51012301705214334157245186176, 3.22887150114067356712133265076, 4.27061044370522814870291213619, 4.61632169200811206064256954699, 6.03642745594146154833359141072, 6.42922593095694567356099724039, 7.24687800066375950313695672025, 7.85829872364055419138274199540