| L(s) = 1 | + 0.735·3-s + 1.46·5-s + 2.78·7-s − 2.45·9-s − 5.24·13-s + 1.07·15-s − 0.661·17-s − 19-s + 2.05·21-s + 3.27·23-s − 2.85·25-s − 4.01·27-s − 5.31·29-s − 4.78·31-s + 4.08·35-s + 2.24·37-s − 3.86·39-s + 5.21·41-s + 7.42·43-s − 3.60·45-s + 13.5·47-s + 0.770·49-s − 0.486·51-s − 9.59·53-s − 0.735·57-s − 5.95·59-s − 10.5·61-s + ⋯ |
| L(s) = 1 | + 0.424·3-s + 0.655·5-s + 1.05·7-s − 0.819·9-s − 1.45·13-s + 0.278·15-s − 0.160·17-s − 0.229·19-s + 0.447·21-s + 0.682·23-s − 0.570·25-s − 0.773·27-s − 0.986·29-s − 0.859·31-s + 0.690·35-s + 0.369·37-s − 0.618·39-s + 0.813·41-s + 1.13·43-s − 0.537·45-s + 1.97·47-s + 0.110·49-s − 0.0681·51-s − 1.31·53-s − 0.0974·57-s − 0.775·59-s − 1.35·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.735T + 3T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 + 5.24T + 13T^{2} \) |
| 17 | \( 1 + 0.661T + 17T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 5.21T + 41T^{2} \) |
| 43 | \( 1 - 7.42T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 59 | \( 1 + 5.95T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−7.62585368769722377593852704151, −6.79445454699648883981520497413, −5.72953753888287474906339742505, −5.47121325645028387488221521227, −4.62626581615087628148827510347, −3.90454718243399294201180386977, −2.72743810738930884698592498140, −2.32141890641987602901562322530, −1.46023770928930068732621050880, 0,
1.46023770928930068732621050880, 2.32141890641987602901562322530, 2.72743810738930884698592498140, 3.90454718243399294201180386977, 4.62626581615087628148827510347, 5.47121325645028387488221521227, 5.72953753888287474906339742505, 6.79445454699648883981520497413, 7.62585368769722377593852704151
