| L(s) = 1 | + 3-s − 0.553·5-s − 2.25·7-s + 9-s − 5.96·11-s − 5.06·13-s − 0.553·15-s + 6.67·17-s − 6.35·19-s − 2.25·21-s − 6.49·23-s − 4.69·25-s + 27-s + 6.89·29-s + 7.49·31-s − 5.96·33-s + 1.24·35-s + 3.64·37-s − 5.06·39-s + 9.03·41-s − 10.3·43-s − 0.553·45-s − 0.815·47-s − 1.93·49-s + 6.67·51-s + 8.57·53-s + 3.30·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.247·5-s − 0.850·7-s + 0.333·9-s − 1.79·11-s − 1.40·13-s − 0.143·15-s + 1.61·17-s − 1.45·19-s − 0.491·21-s − 1.35·23-s − 0.938·25-s + 0.192·27-s + 1.27·29-s + 1.34·31-s − 1.03·33-s + 0.210·35-s + 0.599·37-s − 0.810·39-s + 1.41·41-s − 1.57·43-s − 0.0825·45-s − 0.118·47-s − 0.275·49-s + 0.935·51-s + 1.17·53-s + 0.445·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184679795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184679795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 + 0.553T + 5T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 9.03T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.815T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 4.16T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 5.05T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 5.22T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.068386646159205554446206673563, −7.59638337540723193227271739172, −6.73719385094435406651456058387, −5.95311248657641771797881744890, −5.17519247914326774890505512468, −4.39524340918151508739777062542, −3.54311883244574827880647615211, −2.62329252673685682673839572800, −2.28425408280226462841844260435, −0.51629059200047323192397975089,
0.51629059200047323192397975089, 2.28425408280226462841844260435, 2.62329252673685682673839572800, 3.54311883244574827880647615211, 4.39524340918151508739777062542, 5.17519247914326774890505512468, 5.95311248657641771797881744890, 6.73719385094435406651456058387, 7.59638337540723193227271739172, 8.068386646159205554446206673563
