| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 3.85·7-s − 2.23·8-s − 11-s + 1.76·13-s + 2.38·14-s + 1.85·16-s − 1.61·17-s + 6.70·19-s − 0.618·22-s − 7.09·23-s + 1.09·26-s − 6.23·28-s + 3.61·29-s − 3·31-s + 5.61·32-s − 1.00·34-s + 5.76·37-s + 4.14·38-s + 3·41-s − 6·43-s + 1.61·44-s − 4.38·46-s + 5.94·47-s + 7.85·49-s − 2.85·52-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 1.45·7-s − 0.790·8-s − 0.301·11-s + 0.489·13-s + 0.636·14-s + 0.463·16-s − 0.392·17-s + 1.53·19-s − 0.131·22-s − 1.47·23-s + 0.213·26-s − 1.17·28-s + 0.671·29-s − 0.538·31-s + 0.993·32-s − 0.171·34-s + 0.947·37-s + 0.672·38-s + 0.468·41-s − 0.914·43-s + 0.243·44-s − 0.646·46-s + 0.867·47-s + 1.12·49-s − 0.395·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.109738842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109738842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.854T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 + 0.618T + 97T^{2} \) |
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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
−8.923641811610191952005488595345, −7.894215636803311273179184848065, −7.87291854586124570362650004911, −6.42618521545941063802048059409, −5.54783387358895014371530643234, −4.97635323305849684180720540501, −4.24430699203697829255624211409, −3.41074288090260044902666833996, −2.14779495820259948262395512900, −0.909372251615213281544084534352,
0.909372251615213281544084534352, 2.14779495820259948262395512900, 3.41074288090260044902666833996, 4.24430699203697829255624211409, 4.97635323305849684180720540501, 5.54783387358895014371530643234, 6.42618521545941063802048059409, 7.87291854586124570362650004911, 7.894215636803311273179184848065, 8.923641811610191952005488595345
