L(s) = 1 | + 5.23·2-s + 5.47·3-s − 100.·4-s + 28.6·6-s − 769.·7-s − 1.19e3·8-s − 2.15e3·9-s + 5.35e3·11-s − 550.·12-s − 1.40e4·13-s − 4.03e3·14-s + 6.60e3·16-s − 1.24e4·17-s − 1.12e4·18-s − 5.03e3·19-s − 4.21e3·21-s + 2.80e4·22-s + 7.51e4·23-s − 6.55e3·24-s − 7.35e4·26-s − 2.37e4·27-s + 7.74e4·28-s + 1.95e5·29-s − 9.35e4·31-s + 1.87e5·32-s + 2.93e4·33-s − 6.49e4·34-s + ⋯ |
L(s) = 1 | + 0.462·2-s + 0.117·3-s − 0.785·4-s + 0.0542·6-s − 0.848·7-s − 0.826·8-s − 0.986·9-s + 1.21·11-s − 0.0919·12-s − 1.77·13-s − 0.392·14-s + 0.402·16-s − 0.612·17-s − 0.456·18-s − 0.168·19-s − 0.0993·21-s + 0.561·22-s + 1.28·23-s − 0.0967·24-s − 0.820·26-s − 0.232·27-s + 0.666·28-s + 1.48·29-s − 0.564·31-s + 1.01·32-s + 0.142·33-s − 0.283·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 5.23T + 128T^{2} \) |
| 3 | \( 1 - 5.47T + 2.18e3T^{2} \) |
| 7 | \( 1 + 769.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.40e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.03e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.95e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.61e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.06e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.26e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.57e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.74e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.63e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.01e7T + 8.07e13T^{2} \) |
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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
−14.95937024719211189959730435798, −14.18663853440256213057478113455, −12.88966781965988083640468954127, −11.79467104745418520971479149715, −9.752193847258098494785779734218, −8.748632304742897274543466814240, −6.60748954782632877725163608559, −4.88220612848529370419279847640, −3.12223906792142027095375231614, 0,
3.12223906792142027095375231614, 4.88220612848529370419279847640, 6.60748954782632877725163608559, 8.748632304742897274543466814240, 9.752193847258098494785779734218, 11.79467104745418520971479149715, 12.88966781965988083640468954127, 14.18663853440256213057478113455, 14.95937024719211189959730435798