| L(s) = 1 | − 0.434·2-s − 1.75·3-s − 1.81·4-s + 0.763·6-s + 2.56·7-s + 1.65·8-s + 0.0918·9-s − 2.96·11-s + 3.18·12-s − 5.29·13-s − 1.11·14-s + 2.90·16-s + 2.85·17-s − 0.0398·18-s + 6.43·19-s − 4.50·21-s + 1.28·22-s − 7.60·23-s − 2.91·24-s + 2.29·26-s + 5.11·27-s − 4.63·28-s − 7.01·29-s + 3.60·31-s − 4.57·32-s + 5.22·33-s − 1.24·34-s + ⋯ |
| L(s) = 1 | − 0.307·2-s − 1.01·3-s − 0.905·4-s + 0.311·6-s + 0.967·7-s + 0.585·8-s + 0.0306·9-s − 0.895·11-s + 0.919·12-s − 1.46·13-s − 0.297·14-s + 0.725·16-s + 0.693·17-s − 0.00939·18-s + 1.47·19-s − 0.982·21-s + 0.274·22-s − 1.58·23-s − 0.594·24-s + 0.450·26-s + 0.984·27-s − 0.876·28-s − 1.30·29-s + 0.646·31-s − 0.808·32-s + 0.908·33-s − 0.212·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.434T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 - 6.43T + 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 + 7.01T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 0.899T + 37T^{2} \) |
| 41 | \( 1 - 9.31T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 - 4.29T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−7.74455146638685282514773783673, −7.37975582412835823703055531046, −5.89561246570781637639606837335, −5.59800746443355003562219200962, −4.83995160557072547806209581509, −4.47805796600099786476818163981, −3.25198943498181916968163139404, −2.15480797485359389508836716356, −0.954454872759621969789910202290, 0,
0.954454872759621969789910202290, 2.15480797485359389508836716356, 3.25198943498181916968163139404, 4.47805796600099786476818163981, 4.83995160557072547806209581509, 5.59800746443355003562219200962, 5.89561246570781637639606837335, 7.37975582412835823703055531046, 7.74455146638685282514773783673
