L(s) = 1 | − 1.70·2-s − 3.34·3-s + 0.901·4-s + 5.70·6-s + 4.14·7-s + 1.87·8-s + 8.21·9-s − 3.44·11-s − 3.01·12-s + 2.03·13-s − 7.06·14-s − 4.99·16-s − 2.85·17-s − 13.9·18-s − 5.67·19-s − 13.8·21-s + 5.87·22-s + 7.45·23-s − 6.26·24-s − 3.47·26-s − 17.4·27-s + 3.74·28-s − 9.40·29-s − 4.10·31-s + 4.75·32-s + 11.5·33-s + 4.85·34-s + ⋯ |
L(s) = 1 | − 1.20·2-s − 1.93·3-s + 0.450·4-s + 2.32·6-s + 1.56·7-s + 0.661·8-s + 2.73·9-s − 1.03·11-s − 0.871·12-s + 0.565·13-s − 1.88·14-s − 1.24·16-s − 0.691·17-s − 3.29·18-s − 1.30·19-s − 3.03·21-s + 1.25·22-s + 1.55·23-s − 1.27·24-s − 0.680·26-s − 3.36·27-s + 0.706·28-s − 1.74·29-s − 0.737·31-s + 0.841·32-s + 2.01·33-s + 0.832·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3756800608\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3756800608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 6.62T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 9.69T + 47T^{2} \) |
| 53 | \( 1 - 1.63T + 53T^{2} \) |
| 59 | \( 1 - 0.368T + 59T^{2} \) |
| 61 | \( 1 - 6.74T + 61T^{2} \) |
| 67 | \( 1 + 7.33T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.053505240387103703732713815995, −7.27192711072389087780595766506, −6.98254501834128066575140073557, −5.80442422596947349918537397874, −5.33254617943773221465597887996, −4.61932952405836848093840877012, −4.11810320290503365363223428290, −2.09402265169406140974683099974, −1.45436596384033576905345392292, −0.46410852458013244411186946306,
0.46410852458013244411186946306, 1.45436596384033576905345392292, 2.09402265169406140974683099974, 4.11810320290503365363223428290, 4.61932952405836848093840877012, 5.33254617943773221465597887996, 5.80442422596947349918537397874, 6.98254501834128066575140073557, 7.27192711072389087780595766506, 8.053505240387103703732713815995