| L(s) = 1 | − 0.414·2-s − 3-s − 1.82·4-s − 5-s + 0.414·6-s + 1.82·7-s + 1.58·8-s + 9-s + 0.414·10-s − 2.82·11-s + 1.82·12-s − 1.82·13-s − 0.757·14-s + 15-s + 3·16-s − 1.17·17-s − 0.414·18-s + 1.82·20-s − 1.82·21-s + 1.17·22-s − 0.828·23-s − 1.58·24-s + 25-s + 0.757·26-s − 27-s − 3.34·28-s + 9.65·29-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s + 0.169·6-s + 0.691·7-s + 0.560·8-s + 0.333·9-s + 0.130·10-s − 0.852·11-s + 0.527·12-s − 0.507·13-s − 0.202·14-s + 0.258·15-s + 0.750·16-s − 0.284·17-s − 0.0976·18-s + 0.408·20-s − 0.398·21-s + 0.249·22-s − 0.172·23-s − 0.323·24-s + 0.200·25-s + 0.148·26-s − 0.192·27-s − 0.631·28-s + 1.79·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 9.48T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.87587639425027760913783960660, −7.33464165076550299869042508898, −6.37024367687024369583995842782, −5.42574763819654112849576319027, −4.84403765704057102594125840113, −4.39694039019535529478385784377, −3.39848913040142891201725273066, −2.23558743755179900952322903925, −1.01880189932161573933628501582, 0,
1.01880189932161573933628501582, 2.23558743755179900952322903925, 3.39848913040142891201725273066, 4.39694039019535529478385784377, 4.84403765704057102594125840113, 5.42574763819654112849576319027, 6.37024367687024369583995842782, 7.33464165076550299869042508898, 7.87587639425027760913783960660
