| L(s) = 1 | + 2.54·3-s + 3.81·5-s − 2.11·7-s + 3.46·9-s + 0.101·11-s + 1.74·13-s + 9.69·15-s − 1.86·17-s + 5.94·19-s − 5.37·21-s − 6.27·23-s + 9.56·25-s + 1.17·27-s + 4.57·29-s + 6.24·31-s + 0.257·33-s − 8.06·35-s + 9.81·37-s + 4.42·39-s − 1.83·41-s − 5.43·43-s + 13.2·45-s − 2.44·47-s − 2.53·49-s − 4.72·51-s − 1.44·53-s + 0.387·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.70·5-s − 0.799·7-s + 1.15·9-s + 0.0305·11-s + 0.483·13-s + 2.50·15-s − 0.451·17-s + 1.36·19-s − 1.17·21-s − 1.30·23-s + 1.91·25-s + 0.225·27-s + 0.848·29-s + 1.12·31-s + 0.0448·33-s − 1.36·35-s + 1.61·37-s + 0.709·39-s − 0.286·41-s − 0.829·43-s + 1.96·45-s − 0.356·47-s − 0.361·49-s − 0.662·51-s − 0.198·53-s + 0.0521·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.431559121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.431559121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 - 0.101T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 23 | \( 1 + 6.27T + 23T^{2} \) |
| 29 | \( 1 - 4.57T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 9.81T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 5.43T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 + 2.61T + 83T^{2} \) |
| 89 | \( 1 + 4.93T + 89T^{2} \) |
| 97 | \( 1 + 8.14T + 97T^{2} \) |
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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
−8.444843918318828292168704206936, −8.004575578717307189351213736125, −6.86703927220268171600058319741, −6.31275498249648667890899201697, −5.63095101419219391029178526428, −4.59596860487049156235833209534, −3.52355985932249279385076739262, −2.82450287260689510075323681801, −2.20519658582971134239208108345, −1.23519598544256832702527521976,
1.23519598544256832702527521976, 2.20519658582971134239208108345, 2.82450287260689510075323681801, 3.52355985932249279385076739262, 4.59596860487049156235833209534, 5.63095101419219391029178526428, 6.31275498249648667890899201697, 6.86703927220268171600058319741, 8.004575578717307189351213736125, 8.444843918318828292168704206936
