| L(s) = 1 | + 1.56·3-s − 3.56·5-s − 7-s − 0.561·9-s + 11-s − 5.12·13-s − 5.56·15-s − 2·17-s + 3.12·19-s − 1.56·21-s − 5.56·23-s + 7.68·25-s − 5.56·27-s − 2·29-s − 6.43·31-s + 1.56·33-s + 3.56·35-s + 0.438·37-s − 8·39-s − 10·41-s + 4·43-s + 2·45-s + 10.2·47-s + 49-s − 3.12·51-s + 12.2·53-s − 3.56·55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 1.59·5-s − 0.377·7-s − 0.187·9-s + 0.301·11-s − 1.42·13-s − 1.43·15-s − 0.485·17-s + 0.716·19-s − 0.340·21-s − 1.15·23-s + 1.53·25-s − 1.07·27-s − 0.371·29-s − 1.15·31-s + 0.271·33-s + 0.602·35-s + 0.0720·37-s − 1.28·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.49·47-s + 0.142·49-s − 0.437·51-s + 1.68·53-s − 0.480·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 0.438T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 9.56T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−10.04900861894279974697769488082, −9.165298715982995101076461140387, −8.398420384640503098321040322054, −7.56165567820588145269391053470, −7.05477591295110836533108705515, −5.49624383309977074006990886382, −4.19955819997859575338975976910, −3.49576616605916737303806191153, −2.39887700497965227885451748768, 0,
2.39887700497965227885451748768, 3.49576616605916737303806191153, 4.19955819997859575338975976910, 5.49624383309977074006990886382, 7.05477591295110836533108705515, 7.56165567820588145269391053470, 8.398420384640503098321040322054, 9.165298715982995101076461140387, 10.04900861894279974697769488082
