L(s) = 1 | − 1.56i·2-s − 0.438·4-s + i·7-s − 2.43i·8-s − 11-s + 4.56i·13-s + 1.56·14-s − 4.68·16-s + 5.56i·17-s − 3·19-s + 1.56i·22-s − 2.43i·23-s + 7.12·26-s − 0.438i·28-s + 3.12·29-s + ⋯ |
L(s) = 1 | − 1.10i·2-s − 0.219·4-s + 0.377i·7-s − 0.862i·8-s − 0.301·11-s + 1.26i·13-s + 0.417·14-s − 1.17·16-s + 1.34i·17-s − 0.688·19-s + 0.332i·22-s − 0.508i·23-s + 1.39·26-s − 0.0828i·28-s + 0.579·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.248151717\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248151717\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.56iT - 2T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 13 | \( 1 - 4.56iT - 13T^{2} \) |
| 17 | \( 1 - 5.56iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 2.43iT - 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 - 9.80iT - 37T^{2} \) |
| 41 | \( 1 - 4.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68iT - 43T^{2} \) |
| 47 | \( 1 - 9.56iT - 47T^{2} \) |
| 53 | \( 1 + 7.12iT - 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 - 5.68iT - 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.246iT - 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 + 6.12iT - 97T^{2} \) |
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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
−9.111483726229061210929306256828, −8.475742620599365246801090880536, −7.49345686471001541879329320375, −6.50945385512956203055514212539, −6.06148699124727451510274318033, −4.67080445610505037692380428754, −4.07067266375418368450761482522, −3.06817992036493567550223452121, −2.16484987816460293698095045957, −1.39513690258621475943401545764,
0.39716261025333494909450183199, 2.14094245115857298600516448812, 3.11483625011627001251427772206, 4.27337587773533123366520800000, 5.35165299846590276613134626846, 5.64304869033236795391955112728, 6.70064717801022663665219453387, 7.47588835261568541979098186309, 7.70961719876577766525131831637, 8.761483073963686508256325035752