| L(s) = 1 | + 2-s − 3-s + 4-s − 3.27·5-s − 6-s − 7-s + 8-s + 9-s − 3.27·10-s − 5.27·11-s − 12-s − 14-s + 3.27·15-s + 16-s − 7.27·17-s + 18-s − 5.27·19-s − 3.27·20-s + 21-s − 5.27·22-s − 5.27·23-s − 24-s + 5.72·25-s − 27-s − 28-s + 0.725·29-s + 3.27·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.46·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.03·10-s − 1.59·11-s − 0.288·12-s − 0.267·14-s + 0.845·15-s + 0.250·16-s − 1.76·17-s + 0.235·18-s − 1.21·19-s − 0.732·20-s + 0.218·21-s − 1.12·22-s − 1.09·23-s − 0.204·24-s + 1.14·25-s − 0.192·27-s − 0.188·28-s + 0.134·29-s + 0.597·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3090460083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3090460083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 3.27T + 37T^{2} \) |
| 41 | \( 1 - 8.54T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.66355750417991140697761149300, −7.29520075964807195407607261495, −6.45522411028721007232063303897, −5.81326612013499965031562730859, −4.98440869850844187168806825124, −4.22906947203251665901274850447, −3.93396769669934073899606104063, −2.79288000868637548437647632133, −2.07170418726121768412641619059, −0.24003321774572802507874822758,
0.24003321774572802507874822758, 2.07170418726121768412641619059, 2.79288000868637548437647632133, 3.93396769669934073899606104063, 4.22906947203251665901274850447, 4.98440869850844187168806825124, 5.81326612013499965031562730859, 6.45522411028721007232063303897, 7.29520075964807195407607261495, 7.66355750417991140697761149300
