| L(s) = 1 | − 1.96·3-s + 5-s − 2.16·7-s + 0.851·9-s + 3.01·11-s − 2.71·13-s − 1.96·15-s + 0.748·17-s − 4.29·19-s + 4.25·21-s + 1.71·23-s + 25-s + 4.21·27-s − 1.74·29-s + 4.16·31-s − 5.91·33-s − 2.16·35-s + 4.69·37-s + 5.32·39-s − 9.89·41-s + 8.49·43-s + 0.851·45-s + 12.8·47-s − 2.29·49-s − 1.46·51-s − 7.88·53-s + 3.01·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s + 0.447·5-s − 0.819·7-s + 0.283·9-s + 0.908·11-s − 0.752·13-s − 0.506·15-s + 0.181·17-s − 0.984·19-s + 0.928·21-s + 0.358·23-s + 0.200·25-s + 0.811·27-s − 0.324·29-s + 0.748·31-s − 1.02·33-s − 0.366·35-s + 0.771·37-s + 0.853·39-s − 1.54·41-s + 1.29·43-s + 0.126·45-s + 1.86·47-s − 0.327·49-s − 0.205·51-s − 1.08·53-s + 0.406·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 - 0.748T + 17T^{2} \) |
| 19 | \( 1 + 4.29T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 - 4.69T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 7.88T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 + 6.64T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 + 5.44T + 79T^{2} \) |
| 83 | \( 1 - 4.01T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 2.91T + 97T^{2} \) |
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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
−7.48405200601186976402034940732, −6.77002783187613666480984446487, −6.22662497464482199215854512205, −5.79354016972881411883571329411, −4.88635450053561325510248725621, −4.24841509730646980342919338533, −3.20214529425128953804875903106, −2.30612612059409720093110126665, −1.08895334580445830326054490229, 0,
1.08895334580445830326054490229, 2.30612612059409720093110126665, 3.20214529425128953804875903106, 4.24841509730646980342919338533, 4.88635450053561325510248725621, 5.79354016972881411883571329411, 6.22662497464482199215854512205, 6.77002783187613666480984446487, 7.48405200601186976402034940732
