| L(s) = 1 | + 2.39·2-s + 3-s + 3.75·4-s + 2.39·6-s − 2.59·7-s + 4.20·8-s + 9-s − 2.31·11-s + 3.75·12-s + 0.510·13-s − 6.23·14-s + 2.57·16-s + 1.70·17-s + 2.39·18-s + 7.72·19-s − 2.59·21-s − 5.56·22-s − 0.207·23-s + 4.20·24-s + 1.22·26-s + 27-s − 9.75·28-s + 4.79·29-s + 8.71·31-s − 2.22·32-s − 2.31·33-s + 4.09·34-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.577·3-s + 1.87·4-s + 0.979·6-s − 0.982·7-s + 1.48·8-s + 0.333·9-s − 0.699·11-s + 1.08·12-s + 0.141·13-s − 1.66·14-s + 0.643·16-s + 0.413·17-s + 0.565·18-s + 1.77·19-s − 0.567·21-s − 1.18·22-s − 0.0431·23-s + 0.857·24-s + 0.240·26-s + 0.192·27-s − 1.84·28-s + 0.890·29-s + 1.56·31-s − 0.394·32-s − 0.403·33-s + 0.701·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.957906582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.957906582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 13 | \( 1 - 0.510T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 - 7.72T + 19T^{2} \) |
| 23 | \( 1 + 0.207T + 23T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 - 8.71T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 6.04T + 41T^{2} \) |
| 43 | \( 1 + 3.98T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 + 0.544T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.943T + 61T^{2} \) |
| 67 | \( 1 - 0.605T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.73340923037053244209429402341, −6.80035950414445800423343228493, −6.42525849738043880454488055185, −5.55609713632232459865476902081, −5.03408111837157209677642878794, −4.22749003946163079933129454020, −3.42848690419933320055227193490, −2.95778406063704145366203515669, −2.39925189673798076485277179735, −0.996773433712542550665219272136,
0.996773433712542550665219272136, 2.39925189673798076485277179735, 2.95778406063704145366203515669, 3.42848690419933320055227193490, 4.22749003946163079933129454020, 5.03408111837157209677642878794, 5.55609713632232459865476902081, 6.42525849738043880454488055185, 6.80035950414445800423343228493, 7.73340923037053244209429402341
