L(s) = 1 | − 1.34·3-s + 0.0749·5-s − 4.25·7-s − 1.18·9-s + 1.12·11-s − 0.485·13-s − 0.100·15-s + 6.54·17-s − 3.86·19-s + 5.72·21-s − 8.49·23-s − 4.99·25-s + 5.63·27-s − 1.99·29-s − 6.58·31-s − 1.51·33-s − 0.318·35-s + 5.71·37-s + 0.654·39-s + 4.59·41-s − 6.88·43-s − 0.0887·45-s + 2.43·47-s + 11.0·49-s − 8.81·51-s − 10.3·53-s + 0.0840·55-s + ⋯ |
L(s) = 1 | − 0.777·3-s + 0.0334·5-s − 1.60·7-s − 0.394·9-s + 0.338·11-s − 0.134·13-s − 0.0260·15-s + 1.58·17-s − 0.886·19-s + 1.25·21-s − 1.77·23-s − 0.998·25-s + 1.08·27-s − 0.371·29-s − 1.18·31-s − 0.263·33-s − 0.0538·35-s + 0.939·37-s + 0.104·39-s + 0.717·41-s − 1.05·43-s − 0.0132·45-s + 0.355·47-s + 1.58·49-s − 1.23·51-s − 1.42·53-s + 0.0113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4438949164\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4438949164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 - 0.0749T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + 0.485T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 8.49T + 23T^{2} \) |
| 29 | \( 1 + 1.99T + 29T^{2} \) |
| 31 | \( 1 + 6.58T + 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 + 6.88T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 + 5.48T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.951349624332411554869625743361, −7.33909748494736757747334160509, −6.31120100909483601731431718520, −6.04156849273486791924642614436, −5.55648304038563484405130903655, −4.38137145247104761035350467970, −3.60946202518809770658705053761, −2.94599164104428814386303528833, −1.78898968257527753857284283055, −0.34923828408716883700221387056,
0.34923828408716883700221387056, 1.78898968257527753857284283055, 2.94599164104428814386303528833, 3.60946202518809770658705053761, 4.38137145247104761035350467970, 5.55648304038563484405130903655, 6.04156849273486791924642614436, 6.31120100909483601731431718520, 7.33909748494736757747334160509, 7.951349624332411554869625743361