| L(s) = 1 | + 2.59·3-s − 1.92·5-s + 1.80·7-s + 3.70·9-s − 1.71·11-s + 0.374·13-s − 4.99·15-s − 4.23·17-s − 7.16·19-s + 4.67·21-s − 1.54·23-s − 1.27·25-s + 1.83·27-s − 3.54·29-s − 9.29·31-s − 4.45·33-s − 3.48·35-s + 0.384·37-s + 0.968·39-s + 4.15·41-s − 2.72·43-s − 7.15·45-s + 0.802·47-s − 3.74·49-s − 10.9·51-s + 2.16·53-s + 3.31·55-s + ⋯ |
| L(s) = 1 | + 1.49·3-s − 0.863·5-s + 0.682·7-s + 1.23·9-s − 0.518·11-s + 0.103·13-s − 1.29·15-s − 1.02·17-s − 1.64·19-s + 1.02·21-s − 0.321·23-s − 0.255·25-s + 0.352·27-s − 0.657·29-s − 1.66·31-s − 0.775·33-s − 0.588·35-s + 0.0631·37-s + 0.155·39-s + 0.649·41-s − 0.415·43-s − 1.06·45-s + 0.117·47-s − 0.534·49-s − 1.53·51-s + 0.296·53-s + 0.447·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.59T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 - 0.374T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 7.16T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 37 | \( 1 - 0.384T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 + 2.72T + 43T^{2} \) |
| 47 | \( 1 - 0.802T + 47T^{2} \) |
| 53 | \( 1 - 2.16T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 - 7.85T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 1.53T + 79T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 + 8.62T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.150444129502074131091432518172, −7.61155095475021714450146178620, −6.94983147318558722767197372554, −5.87751835671902663515477607709, −4.75639692106146797807679988213, −4.05423389719972608592713439680, −3.52267986444287483957580007714, −2.35817137180055013603923168171, −1.87768551529521598739930767093, 0,
1.87768551529521598739930767093, 2.35817137180055013603923168171, 3.52267986444287483957580007714, 4.05423389719972608592713439680, 4.75639692106146797807679988213, 5.87751835671902663515477607709, 6.94983147318558722767197372554, 7.61155095475021714450146178620, 8.150444129502074131091432518172
