| L(s) = 1 | − 1.64·2-s − 3-s + 0.711·4-s + 5-s + 1.64·6-s + 4.47·7-s + 2.12·8-s + 9-s − 1.64·10-s − 3.44·11-s − 0.711·12-s + 2.47·13-s − 7.37·14-s − 15-s − 4.91·16-s + 7.62·17-s − 1.64·18-s + 0.711·20-s − 4.47·21-s + 5.66·22-s + 3.87·23-s − 2.12·24-s + 25-s − 4.08·26-s − 27-s + 3.18·28-s − 8.73·29-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.577·3-s + 0.355·4-s + 0.447·5-s + 0.672·6-s + 1.69·7-s + 0.750·8-s + 0.333·9-s − 0.520·10-s − 1.03·11-s − 0.205·12-s + 0.687·13-s − 1.97·14-s − 0.258·15-s − 1.22·16-s + 1.85·17-s − 0.388·18-s + 0.159·20-s − 0.977·21-s + 1.20·22-s + 0.807·23-s − 0.433·24-s + 0.200·25-s − 0.800·26-s − 0.192·27-s + 0.602·28-s − 1.62·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236304121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236304121\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 - 7.62T + 17T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 + 0.422T + 31T^{2} \) |
| 37 | \( 1 - 3.90T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 0.677T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 - 8.20T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 + 3.85T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.03T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.984230579229777672860831206542, −7.77895635659270776207249189197, −7.12183178301122368234109386896, −5.83413460254597192591443088905, −5.35933199261701773657843743827, −4.76260406093677544437606991414, −3.77049427366780039218381706256, −2.37985640751655054552666070875, −1.46778104332875294373576663589, −0.836410074737917473388168748558,
0.836410074737917473388168748558, 1.46778104332875294373576663589, 2.37985640751655054552666070875, 3.77049427366780039218381706256, 4.76260406093677544437606991414, 5.35933199261701773657843743827, 5.83413460254597192591443088905, 7.12183178301122368234109386896, 7.77895635659270776207249189197, 7.984230579229777672860831206542
