| L(s) = 1 | − 2.55·3-s + 3.69·5-s + 1.87·7-s + 3.51·9-s + 3.15·11-s − 4.32·13-s − 9.43·15-s − 3.84·17-s − 19-s − 4.79·21-s + 5.89·23-s + 8.65·25-s − 1.32·27-s + 2.10·29-s − 6.48·31-s − 8.05·33-s + 6.94·35-s − 1.37·37-s + 11.0·39-s + 11.0·41-s + 4.81·43-s + 13.0·45-s + 11.3·47-s − 3.47·49-s + 9.80·51-s − 53-s + 11.6·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 1.65·5-s + 0.709·7-s + 1.17·9-s + 0.950·11-s − 1.19·13-s − 2.43·15-s − 0.931·17-s − 0.229·19-s − 1.04·21-s + 1.22·23-s + 1.73·25-s − 0.255·27-s + 0.391·29-s − 1.16·31-s − 1.40·33-s + 1.17·35-s − 0.225·37-s + 1.76·39-s + 1.73·41-s + 0.733·43-s + 1.93·45-s + 1.65·47-s − 0.496·49-s + 1.37·51-s − 0.137·53-s + 1.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727462331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727462331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 2.55T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 + 4.32T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 23 | \( 1 - 5.89T + 23T^{2} \) |
| 29 | \( 1 - 2.10T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 4.81T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 3.01T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 4.04T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−8.729726685299385216807697908761, −7.27275881092885655052585142249, −6.88333535845172469422132229797, −6.03136652190204951164748504826, −5.59028165092462545794242613705, −4.87330014779580029775034585244, −4.29989500591035777192419919099, −2.62700327289618778450347627046, −1.81005784935439400989643901801, −0.844004475265707502145530537574,
0.844004475265707502145530537574, 1.81005784935439400989643901801, 2.62700327289618778450347627046, 4.29989500591035777192419919099, 4.87330014779580029775034585244, 5.59028165092462545794242613705, 6.03136652190204951164748504826, 6.88333535845172469422132229797, 7.27275881092885655052585142249, 8.729726685299385216807697908761
