| L(s) = 1 | + 2.37·2-s − 3-s + 3.66·4-s − 2.87·5-s − 2.37·6-s − 7-s + 3.95·8-s + 9-s − 6.83·10-s − 0.240·11-s − 3.66·12-s + 7.14·13-s − 2.37·14-s + 2.87·15-s + 2.08·16-s − 1.45·17-s + 2.37·18-s + 0.0789·19-s − 10.5·20-s + 21-s − 0.572·22-s − 4.83·23-s − 3.95·24-s + 3.25·25-s + 17.0·26-s − 27-s − 3.66·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.83·4-s − 1.28·5-s − 0.971·6-s − 0.377·7-s + 1.39·8-s + 0.333·9-s − 2.16·10-s − 0.0725·11-s − 1.05·12-s + 1.98·13-s − 0.635·14-s + 0.741·15-s + 0.522·16-s − 0.352·17-s + 0.560·18-s + 0.0181·19-s − 2.35·20-s + 0.218·21-s − 0.122·22-s − 1.00·23-s − 0.807·24-s + 0.650·25-s + 3.33·26-s − 0.192·27-s − 0.692·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.421858467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421858467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 11 | \( 1 + 0.240T + 11T^{2} \) |
| 13 | \( 1 - 7.14T + 13T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 - 0.0789T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 0.514T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 0.826T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 9.24T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 3.00T + 73T^{2} \) |
| 79 | \( 1 + 2.81T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.71920459278173833854011350583, −6.65059907480159599025097632372, −6.36994139105112024910230734771, −5.75403182030650307001019122133, −4.86775061097194241284257961817, −4.21663923196402117626163386740, −3.71101197690309483752468196880, −3.20977755486470385124209594298, −2.03561504111757221260712933573, −0.72082837890281645811217038591,
0.72082837890281645811217038591, 2.03561504111757221260712933573, 3.20977755486470385124209594298, 3.71101197690309483752468196880, 4.21663923196402117626163386740, 4.86775061097194241284257961817, 5.75403182030650307001019122133, 6.36994139105112024910230734771, 6.65059907480159599025097632372, 7.71920459278173833854011350583
