| L(s) = 1 | − 3-s + 0.957·7-s + 9-s − 5.41·11-s + 2.02·13-s + 0.642·17-s − 5.04·19-s − 0.957·21-s − 3.51·23-s − 27-s + 10.1·29-s + 3.69·31-s + 5.41·33-s + 11.3·37-s − 2.02·39-s + 3.52·41-s − 0.766·43-s − 4.93·47-s − 6.08·49-s − 0.642·51-s − 5.94·53-s + 5.04·57-s + 4.71·59-s + 4.34·61-s + 0.957·63-s − 9.51·67-s + 3.51·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.361·7-s + 0.333·9-s − 1.63·11-s + 0.560·13-s + 0.155·17-s − 1.15·19-s − 0.208·21-s − 0.733·23-s − 0.192·27-s + 1.88·29-s + 0.663·31-s + 0.942·33-s + 1.86·37-s − 0.323·39-s + 0.550·41-s − 0.116·43-s − 0.719·47-s − 0.869·49-s − 0.0900·51-s − 0.816·53-s + 0.667·57-s + 0.614·59-s + 0.556·61-s + 0.120·63-s − 1.16·67-s + 0.423·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.957T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 - 0.642T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 + 3.51T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 0.766T + 43T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.39T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.80286735530267402182299745319, −6.62615573690272750120000376146, −6.20356965530777544426741663435, −5.44030292265310701744594887587, −4.69250380512107642497467904851, −4.21033141066025152963866794507, −2.97957903960352975459875026411, −2.31227836518603989065416147418, −1.15640529661877207903229062742, 0,
1.15640529661877207903229062742, 2.31227836518603989065416147418, 2.97957903960352975459875026411, 4.21033141066025152963866794507, 4.69250380512107642497467904851, 5.44030292265310701744594887587, 6.20356965530777544426741663435, 6.62615573690272750120000376146, 7.80286735530267402182299745319
