L(s) = 1 | − 176.·2-s − 1.55e3·3-s + 1.48e4·4-s + 2.86e4i·5-s + 2.74e5·6-s + 1.02e6i·7-s + 2.66e5·8-s − 2.36e6·9-s − 5.05e6i·10-s − 7.94e6i·11-s − 2.31e7·12-s + 7.50e7·13-s − 1.80e8i·14-s − 4.45e7i·15-s − 2.90e8·16-s − 3.73e8i·17-s + ⋯ |
L(s) = 1 | − 1.38·2-s − 0.711·3-s + 0.907·4-s + 0.366i·5-s + 0.982·6-s + 1.24i·7-s + 0.127·8-s − 0.494·9-s − 0.505i·10-s − 0.407i·11-s − 0.645·12-s + 1.19·13-s − 1.71i·14-s − 0.260i·15-s − 1.08·16-s − 0.910i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00196 - 0.999i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.00196 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.5201899898\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5201899898\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (6.69e6 - 3.40e9i)T \) |
good | 2 | \( 1 + 176.T + 1.63e4T^{2} \) |
| 3 | \( 1 + 1.55e3T + 4.78e6T^{2} \) |
| 5 | \( 1 - 2.86e4iT - 6.10e9T^{2} \) |
| 7 | \( 1 - 1.02e6iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 7.94e6iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 7.50e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 3.73e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.47e9iT - 7.99e17T^{2} \) |
| 29 | \( 1 + 2.04e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 1.05e10T + 7.56e20T^{2} \) |
| 37 | \( 1 - 1.71e11iT - 9.01e21T^{2} \) |
| 41 | \( 1 - 1.96e11T + 3.79e22T^{2} \) |
| 43 | \( 1 + 1.63e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 - 1.29e11T + 2.56e23T^{2} \) |
| 53 | \( 1 + 1.49e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 2.16e11T + 6.19e24T^{2} \) |
| 61 | \( 1 - 1.53e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 2.75e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 - 8.43e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + 1.34e13T + 1.22e26T^{2} \) |
| 79 | \( 1 - 3.08e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 - 1.79e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 2.82e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 1.46e14iT - 6.52e27T^{2} \) |
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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
−15.38497146528021665624884681510, −13.52349296776270233054029278218, −11.56281758810526041042810962720, −11.02238730642551709206488743449, −9.322131215248779472031457894536, −8.470045787149720590509823160076, −6.74150931349793108355562859974, −5.35085317907718175850054861385, −2.70193970106383696577912370832, −0.891483073013022757782725018301,
0.42985690112440866944188395007, 1.45029874275154884542109969068, 4.13253692898809722295574821531, 6.11406876745030788875703685020, 7.65275150143523508273407972320, 8.814075772481818182985054441673, 10.38993089796251741115018601020, 10.98711670801094566409577184410, 12.71533129818285643637296299111, 14.26777917957455041788809401628