L(s) = 1 | + 3-s − 3.94·5-s − 7-s + 9-s − 1.72·11-s + 2.21·13-s − 3.94·15-s − 5.14·17-s + 1.72·19-s − 21-s + 23-s + 10.5·25-s + 27-s + 3.01·29-s + 8.20·31-s − 1.72·33-s + 3.94·35-s + 7.61·37-s + 2.21·39-s + 1.14·41-s − 7.22·43-s − 3.94·45-s − 7.34·47-s + 49-s − 5.14·51-s − 4.21·53-s + 6.82·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.76·5-s − 0.377·7-s + 0.333·9-s − 0.521·11-s + 0.614·13-s − 1.01·15-s − 1.24·17-s + 0.396·19-s − 0.218·21-s + 0.208·23-s + 2.11·25-s + 0.192·27-s + 0.558·29-s + 1.47·31-s − 0.301·33-s + 0.666·35-s + 1.25·37-s + 0.354·39-s + 0.179·41-s − 1.10·43-s − 0.588·45-s − 1.07·47-s + 0.142·49-s − 0.720·51-s − 0.578·53-s + 0.919·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3.94T + 5T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 29 | \( 1 - 3.01T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 4.21T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 0.0825T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 + 8.33T + 73T^{2} \) |
| 79 | \( 1 + 8.20T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 9.45T + 89T^{2} \) |
| 97 | \( 1 + 8.33T + 97T^{2} \) |
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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
−7.75319712571790788886950948943, −6.79150490468149509658498424893, −6.51396687852448522575213573882, −5.20131257236212580212902161350, −4.43846301463721440147027728352, −3.94739389278496644838021107248, −3.12737881616901594223540217925, −2.56488344005516336837722214407, −1.09889441847995240216728185159, 0,
1.09889441847995240216728185159, 2.56488344005516336837722214407, 3.12737881616901594223540217925, 3.94739389278496644838021107248, 4.43846301463721440147027728352, 5.20131257236212580212902161350, 6.51396687852448522575213573882, 6.79150490468149509658498424893, 7.75319712571790788886950948943