| L(s) = 1 | − 2.48i·3-s + i·5-s − 3.58·7-s − 3.19·9-s − 4.71i·11-s + 0.957i·13-s + 2.48·15-s − 1.56·17-s + i·19-s + 8.92i·21-s + 2.80·23-s − 25-s + 0.475i·27-s + 5.01i·29-s − 7.59·31-s + ⋯ |
| L(s) = 1 | − 1.43i·3-s + 0.447i·5-s − 1.35·7-s − 1.06·9-s − 1.42i·11-s + 0.265i·13-s + 0.642·15-s − 0.380·17-s + 0.229i·19-s + 1.94i·21-s + 0.584·23-s − 0.200·25-s + 0.0914i·27-s + 0.931i·29-s − 1.36·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3198701804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3198701804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 + 2.48iT - 3T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 4.71iT - 11T^{2} \) |
| 13 | \( 1 - 0.957iT - 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 23 | \( 1 - 2.80T + 23T^{2} \) |
| 29 | \( 1 - 5.01iT - 29T^{2} \) |
| 31 | \( 1 + 7.59T + 31T^{2} \) |
| 37 | \( 1 + 5.75iT - 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 - 2.12iT - 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 + 0.106iT - 53T^{2} \) |
| 59 | \( 1 - 13.4iT - 59T^{2} \) |
| 61 | \( 1 - 5.34iT - 61T^{2} \) |
| 67 | \( 1 - 6.87iT - 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 9.57iT - 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 7.85T + 97T^{2} \) |
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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
−8.825418479935009069205011092140, −7.983062151781617226862363093465, −7.11440937093689015645787474738, −6.72084098730635242104033195585, −6.07509304400320668178061492229, −5.39948560585971291768165645117, −3.78443429409317958024786209850, −3.14164710429558548511299078618, −2.27517956167084614431922414414, −1.04902006034451765988178825937,
0.10999631613795843799153909006, 2.02962999103101686974848829208, 3.23941114765170429460692012466, 3.81635915918968048371936562548, 4.76761084391335443884047511868, 5.16132981357008729067517128952, 6.30301392554650349770524710098, 6.95668546328314901977986479312, 7.929812517828139814120285266227, 8.970819983451015950261269749305
