| L(s) = 1 | + 2.14·2-s + 2.59·4-s + 0.447·5-s + 4.87·7-s + 1.26·8-s + 0.958·10-s + 11-s − 13-s + 10.4·14-s − 2.47·16-s + 6.61·17-s − 5.98·19-s + 1.15·20-s + 2.14·22-s − 0.694·23-s − 4.79·25-s − 2.14·26-s + 12.6·28-s + 2.14·29-s − 2.44·31-s − 7.82·32-s + 14.1·34-s + 2.18·35-s + 0.921·37-s − 12.8·38-s + 0.565·40-s + 1.12·41-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.29·4-s + 0.200·5-s + 1.84·7-s + 0.446·8-s + 0.303·10-s + 0.301·11-s − 0.277·13-s + 2.79·14-s − 0.617·16-s + 1.60·17-s − 1.37·19-s + 0.259·20-s + 0.456·22-s − 0.144·23-s − 0.959·25-s − 0.420·26-s + 2.38·28-s + 0.398·29-s − 0.439·31-s − 1.38·32-s + 2.43·34-s + 0.369·35-s + 0.151·37-s − 2.07·38-s + 0.0894·40-s + 0.175·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.570059725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.570059725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 - 0.447T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 + 5.98T + 19T^{2} \) |
| 23 | \( 1 + 0.694T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 - 0.921T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 + 5.01T + 53T^{2} \) |
| 59 | \( 1 - 7.47T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 - 1.77T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 4.57T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−9.810032190346759108361184736982, −8.652465390692577185468396059777, −7.915814202069602045391538304335, −7.06599160536066155464213100263, −5.88763771698138838503948072776, −5.40787552909878493012094917774, −4.49083878357832866358736396577, −3.91280817727259572340131972978, −2.56719054574827092010596265068, −1.57919185898740780742217845838,
1.57919185898740780742217845838, 2.56719054574827092010596265068, 3.91280817727259572340131972978, 4.49083878357832866358736396577, 5.40787552909878493012094917774, 5.88763771698138838503948072776, 7.06599160536066155464213100263, 7.915814202069602045391538304335, 8.652465390692577185468396059777, 9.810032190346759108361184736982
