| L(s) = 1 | − 8.38e6·2-s + 2.70e11·3-s + 7.03e13·4-s − 1.02e16·5-s − 2.27e18·6-s + 1.08e20·7-s − 5.90e20·8-s + 4.68e22·9-s + 8.60e22·10-s + 3.92e24·11-s + 1.90e25·12-s − 2.33e26·13-s − 9.13e26·14-s − 2.77e27·15-s + 4.95e27·16-s + 4.05e28·17-s − 3.92e29·18-s − 1.56e29·19-s − 7.21e29·20-s + 2.94e31·21-s − 3.29e31·22-s + 3.32e31·23-s − 1.59e32·24-s − 6.05e32·25-s + 1.95e33·26-s + 5.48e33·27-s + 7.65e33·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s − 0.384·5-s − 1.17·6-s + 1.50·7-s − 0.353·8-s + 1.76·9-s + 0.271·10-s + 1.32·11-s + 0.830·12-s − 1.55·13-s − 1.06·14-s − 0.639·15-s + 0.250·16-s + 0.492·17-s − 1.24·18-s − 0.139·19-s − 0.192·20-s + 2.49·21-s − 0.935·22-s + 0.332·23-s − 0.587·24-s − 0.852·25-s + 1.09·26-s + 1.26·27-s + 0.751·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(\approx\) |
\(2.975398410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.975398410\) |
| \(L(\frac{49}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.38e6T \) |
| good | 3 | \( 1 - 2.70e11T + 2.65e22T^{2} \) |
| 5 | \( 1 + 1.02e16T + 7.10e32T^{2} \) |
| 7 | \( 1 - 1.08e20T + 5.24e39T^{2} \) |
| 11 | \( 1 - 3.92e24T + 8.81e48T^{2} \) |
| 13 | \( 1 + 2.33e26T + 2.26e52T^{2} \) |
| 17 | \( 1 - 4.05e28T + 6.77e57T^{2} \) |
| 19 | \( 1 + 1.56e29T + 1.26e60T^{2} \) |
| 23 | \( 1 - 3.32e31T + 1.00e64T^{2} \) |
| 29 | \( 1 - 9.92e32T + 5.40e68T^{2} \) |
| 31 | \( 1 - 1.71e35T + 1.24e70T^{2} \) |
| 37 | \( 1 - 1.08e37T + 5.07e73T^{2} \) |
| 41 | \( 1 + 7.95e37T + 6.32e75T^{2} \) |
| 43 | \( 1 - 3.70e38T + 5.92e76T^{2} \) |
| 47 | \( 1 + 6.30e38T + 3.87e78T^{2} \) |
| 53 | \( 1 - 2.32e40T + 1.09e81T^{2} \) |
| 59 | \( 1 + 3.80e41T + 1.69e83T^{2} \) |
| 61 | \( 1 - 1.07e42T + 8.13e83T^{2} \) |
| 67 | \( 1 + 1.46e42T + 6.69e85T^{2} \) |
| 71 | \( 1 + 1.66e43T + 1.02e87T^{2} \) |
| 73 | \( 1 + 5.99e42T + 3.76e87T^{2} \) |
| 79 | \( 1 + 1.80e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 2.37e45T + 1.57e90T^{2} \) |
| 89 | \( 1 - 1.07e45T + 4.18e91T^{2} \) |
| 97 | \( 1 + 5.98e45T + 2.38e93T^{2} \) |
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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
−17.30869601008150832221953436288, −15.03254116980632639876689598475, −14.27660402495919206933351349254, −11.81217559882232275802010381244, −9.661025476722952792623206304555, −8.349471305303799656929193553936, −7.42408072172744007028117126382, −4.28534652448586902058129702984, −2.52476585088168058541224491049, −1.30729127808266273151362806314,
1.30729127808266273151362806314, 2.52476585088168058541224491049, 4.28534652448586902058129702984, 7.42408072172744007028117126382, 8.349471305303799656929193553936, 9.661025476722952792623206304555, 11.81217559882232275802010381244, 14.27660402495919206933351349254, 15.03254116980632639876689598475, 17.30869601008150832221953436288
