| L(s) = 1 | − 3·3-s + 10·5-s + 8·7-s + 9·9-s − 40·11-s + 13·13-s − 30·15-s + 130·17-s + 20·19-s − 24·21-s − 25·25-s − 27·27-s − 18·29-s + 184·31-s + 120·33-s + 80·35-s − 74·37-s − 39·39-s − 362·41-s − 76·43-s + 90·45-s + 452·47-s − 279·49-s − 390·51-s + 382·53-s − 400·55-s − 60·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.431·7-s + 1/3·9-s − 1.09·11-s + 0.277·13-s − 0.516·15-s + 1.85·17-s + 0.241·19-s − 0.249·21-s − 1/5·25-s − 0.192·27-s − 0.115·29-s + 1.06·31-s + 0.633·33-s + 0.386·35-s − 0.328·37-s − 0.160·39-s − 1.37·41-s − 0.269·43-s + 0.298·45-s + 1.40·47-s − 0.813·49-s − 1.07·51-s + 0.990·53-s − 0.980·55-s − 0.139·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.047161368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047161368\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 17 | \( 1 - 130 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 452 T + p^{3} T^{2} \) |
| 53 | \( 1 - 382 T + p^{3} T^{2} \) |
| 59 | \( 1 + 464 T + p^{3} T^{2} \) |
| 61 | \( 1 - 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 700 T + p^{3} T^{2} \) |
| 71 | \( 1 - 748 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1058 T + p^{3} T^{2} \) |
| 79 | \( 1 - 976 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 + 386 T + p^{3} T^{2} \) |
| 97 | \( 1 + 614 T + p^{3} T^{2} \) |
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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
−10.15712252291460995872889506849, −9.673468739143442438116269464784, −8.327833090396431167450652898692, −7.61669730631711426130719313272, −6.44816358013138193383018635952, −5.49667370423669418665779992666, −5.05087233816204429818079691917, −3.49052687089720948149207630921, −2.14821524146509874272909847231, −0.901128332814659712947253918340,
0.901128332814659712947253918340, 2.14821524146509874272909847231, 3.49052687089720948149207630921, 5.05087233816204429818079691917, 5.49667370423669418665779992666, 6.44816358013138193383018635952, 7.61669730631711426130719313272, 8.327833090396431167450652898692, 9.673468739143442438116269464784, 10.15712252291460995872889506849
