| L(s) = 1 | + 3-s − 2.96·5-s − 0.804·7-s + 9-s + 2.20·11-s − 0.207·13-s − 2.96·15-s + 7.18·17-s + 6.36·19-s − 0.804·21-s − 0.813·23-s + 3.80·25-s + 27-s − 3.78·29-s + 10.1·31-s + 2.20·33-s + 2.38·35-s + 8.58·37-s − 0.207·39-s − 3.58·41-s + 8.53·43-s − 2.96·45-s − 11.5·47-s − 6.35·49-s + 7.18·51-s + 0.760·53-s − 6.54·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.32·5-s − 0.304·7-s + 0.333·9-s + 0.665·11-s − 0.0574·13-s − 0.766·15-s + 1.74·17-s + 1.45·19-s − 0.175·21-s − 0.169·23-s + 0.760·25-s + 0.192·27-s − 0.702·29-s + 1.82·31-s + 0.384·33-s + 0.403·35-s + 1.41·37-s − 0.0331·39-s − 0.559·41-s + 1.30·43-s − 0.442·45-s − 1.69·47-s − 0.907·49-s + 1.00·51-s + 0.104·53-s − 0.883·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584227754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584227754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 + 0.804T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 0.207T + 13T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 + 0.813T + 23T^{2} \) |
| 29 | \( 1 + 3.78T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 8.58T + 37T^{2} \) |
| 41 | \( 1 + 3.58T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 0.760T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 - 0.443T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 4.94T + 97T^{2} \) |
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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
−9.964337285448155364166899957684, −9.553001771927608417689840708898, −8.327865679226508666912012398354, −7.79496310170491505252404662539, −7.09136201344124884301799008179, −5.89699585919009674172690492448, −4.63762632841431908586469060012, −3.64816454648766133126782610335, −3.01361664001504053558423239498, −1.07257762544020371814181271002,
1.07257762544020371814181271002, 3.01361664001504053558423239498, 3.64816454648766133126782610335, 4.63762632841431908586469060012, 5.89699585919009674172690492448, 7.09136201344124884301799008179, 7.79496310170491505252404662539, 8.327865679226508666912012398354, 9.553001771927608417689840708898, 9.964337285448155364166899957684
