| L(s) = 1 | + 1.94·2-s − 1.97·3-s + 1.76·4-s − 3.82·6-s + 2.17·7-s − 0.458·8-s + 0.888·9-s − 4.36·11-s − 3.47·12-s + 1.66·13-s + 4.22·14-s − 4.41·16-s − 0.0716·17-s + 1.72·18-s + 1.98·19-s − 4.29·21-s − 8.46·22-s + 7.31·23-s + 0.903·24-s + 3.23·26-s + 4.16·27-s + 3.84·28-s + 4.53·29-s − 4.85·31-s − 7.65·32-s + 8.60·33-s − 0.138·34-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 1.13·3-s + 0.881·4-s − 1.56·6-s + 0.822·7-s − 0.162·8-s + 0.296·9-s − 1.31·11-s − 1.00·12-s + 0.461·13-s + 1.12·14-s − 1.10·16-s − 0.0173·17-s + 0.406·18-s + 0.454·19-s − 0.937·21-s − 1.80·22-s + 1.52·23-s + 0.184·24-s + 0.633·26-s + 0.801·27-s + 0.725·28-s + 0.841·29-s − 0.871·31-s − 1.35·32-s + 1.49·33-s − 0.0238·34-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 | \( 1 - 1.94T + 2T^{2} \) |
| 3 | \( 1 + 1.97T + 3T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 + 0.0716T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 - 4.53T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 - 0.626T + 37T^{2} \) |
| 41 | \( 1 - 0.630T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 + 6.53T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 - 0.998T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 - 0.437T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 - 0.0176T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 + 9.52T + 97T^{2} \) |
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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
−7.47629595989574945204748234840, −6.68788093778323192054132248091, −6.03131717868469390842283343272, −5.25719116593342713994981825530, −5.06554273871829156453574866167, −4.43010872533699865437034154328, −3.29083529071132967023123258021, −2.67382063523303523056427730712, −1.36777097801000559693274044868, 0,
1.36777097801000559693274044868, 2.67382063523303523056427730712, 3.29083529071132967023123258021, 4.43010872533699865437034154328, 5.06554273871829156453574866167, 5.25719116593342713994981825530, 6.03131717868469390842283343272, 6.68788093778323192054132248091, 7.47629595989574945204748234840
