| L(s) = 1 | + 289.·2-s − 6.56e3·3-s − 4.73e4·4-s − 1.89e6·6-s − 8.64e6·7-s − 5.16e7·8-s + 4.30e7·9-s + 5.16e8·11-s + 3.10e8·12-s + 5.02e8·13-s − 2.50e9·14-s − 8.72e9·16-s + 9.05e9·17-s + 1.24e10·18-s − 5.98e10·19-s + 5.67e10·21-s + 1.49e11·22-s + 6.13e11·23-s + 3.38e11·24-s + 1.45e11·26-s − 2.82e11·27-s + 4.09e11·28-s + 1.48e12·29-s + 1.99e12·31-s + 4.24e12·32-s − 3.39e12·33-s + 2.61e12·34-s + ⋯ |
| L(s) = 1 | + 0.798·2-s − 0.577·3-s − 0.361·4-s − 0.461·6-s − 0.567·7-s − 1.08·8-s + 0.333·9-s + 0.727·11-s + 0.208·12-s + 0.170·13-s − 0.453·14-s − 0.507·16-s + 0.314·17-s + 0.266·18-s − 0.807·19-s + 0.327·21-s + 0.581·22-s + 1.63·23-s + 0.628·24-s + 0.136·26-s − 0.192·27-s + 0.205·28-s + 0.550·29-s + 0.420·31-s + 0.682·32-s − 0.419·33-s + 0.251·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 289.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 8.64e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 5.16e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 5.02e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 9.05e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 5.98e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.13e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 1.48e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 1.99e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.92e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 6.80e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 3.31e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 9.15e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.19e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 8.64e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 4.01e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 5.83e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 4.57e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 7.42e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.21e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.89e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.27e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 8.83e16T + 5.95e33T^{2} \) |
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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
−10.84002721751373837658445143357, −9.581887040380676439425295408418, −8.632999456760749616295324589872, −6.88947355416115796485562580177, −6.03323307334135936329205170186, −4.94739251197472390671815783109, −3.96829296672959477499571160473, −2.90446355601273318167586485993, −1.13575942326025766449234021568, 0,
1.13575942326025766449234021568, 2.90446355601273318167586485993, 3.96829296672959477499571160473, 4.94739251197472390671815783109, 6.03323307334135936329205170186, 6.88947355416115796485562580177, 8.632999456760749616295324589872, 9.581887040380676439425295408418, 10.84002721751373837658445143357
