| L(s) = 1 | − 11.8·2-s + 81·3-s − 370.·4-s − 963.·6-s + 1.09e4·7-s + 1.04e4·8-s + 6.56e3·9-s − 3.94e4·11-s − 3.00e4·12-s + 5.02e4·13-s − 1.30e5·14-s + 6.48e4·16-s − 4.61e5·17-s − 7.80e4·18-s − 3.70e5·19-s + 8.86e5·21-s + 4.69e5·22-s + 2.29e6·23-s + 8.50e5·24-s − 5.97e5·26-s + 5.31e5·27-s − 4.05e6·28-s − 1.28e6·29-s + 6.51e6·31-s − 6.14e6·32-s − 3.19e6·33-s + 5.48e6·34-s + ⋯ |
| L(s) = 1 | − 0.525·2-s + 0.577·3-s − 0.723·4-s − 0.303·6-s + 1.72·7-s + 0.906·8-s + 0.333·9-s − 0.812·11-s − 0.417·12-s + 0.488·13-s − 0.905·14-s + 0.247·16-s − 1.33·17-s − 0.175·18-s − 0.651·19-s + 0.994·21-s + 0.427·22-s + 1.70·23-s + 0.523·24-s − 0.256·26-s + 0.192·27-s − 1.24·28-s − 0.336·29-s + 1.26·31-s − 1.03·32-s − 0.469·33-s + 0.704·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.881689205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881689205\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 11.8T + 512T^{2} \) |
| 7 | \( 1 - 1.09e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.94e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.02e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.61e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.29e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.45e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.34e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.24e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.45e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.49e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.04e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.46e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.72e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.89e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.21e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.61e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.00e9T + 7.60e17T^{2} \) |
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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
−12.95402808083872596611421088841, −11.22261545826500854398427116828, −10.44455496799221552042975421740, −8.849870672939370368966084802636, −8.421263390648970735605666486155, −7.29682230702268748598941651149, −5.13964563865468749886041247245, −4.23581304620465644697533075921, −2.23210501517038047066633690797, −0.907986123055654896529936135141,
0.907986123055654896529936135141, 2.23210501517038047066633690797, 4.23581304620465644697533075921, 5.13964563865468749886041247245, 7.29682230702268748598941651149, 8.421263390648970735605666486155, 8.849870672939370368966084802636, 10.44455496799221552042975421740, 11.22261545826500854398427116828, 12.95402808083872596611421088841
