L(s) = 1 | − 2.10·2-s − 1.55·3-s + 2.43·4-s − 0.464·5-s + 3.27·6-s − 0.781·7-s − 0.926·8-s − 0.578·9-s + 0.978·10-s − 11-s − 3.79·12-s − 6.89·13-s + 1.64·14-s + 0.722·15-s − 2.92·16-s + 4.41·17-s + 1.21·18-s + 6.82·19-s − 1.13·20-s + 1.21·21-s + 2.10·22-s − 0.0938·23-s + 1.44·24-s − 4.78·25-s + 14.5·26-s + 5.56·27-s − 1.90·28-s + ⋯ |
L(s) = 1 | − 1.48·2-s − 0.898·3-s + 1.21·4-s − 0.207·5-s + 1.33·6-s − 0.295·7-s − 0.327·8-s − 0.192·9-s + 0.309·10-s − 0.301·11-s − 1.09·12-s − 1.91·13-s + 0.440·14-s + 0.186·15-s − 0.731·16-s + 1.07·17-s + 0.287·18-s + 1.56·19-s − 0.253·20-s + 0.265·21-s + 0.449·22-s − 0.0195·23-s + 0.294·24-s − 0.956·25-s + 2.84·26-s + 1.07·27-s − 0.360·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1931765716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1931765716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 + 0.464T + 5T^{2} \) |
| 7 | \( 1 + 0.781T + 7T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 0.0938T + 23T^{2} \) |
| 29 | \( 1 - 6.75T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 3.80T + 53T^{2} \) |
| 59 | \( 1 + 0.976T + 59T^{2} \) |
| 61 | \( 1 + 3.58T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 2.15T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
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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
−7.891775571617475677368519702224, −7.56432256108897651343660754090, −6.96960082658566039167644533487, −6.07413478674288249959382818130, −5.23369513811871983741929409031, −4.80237660831053476841172050769, −3.38408292042692404709436838387, −2.55792368017919531971202983591, −1.41637786386915938863599316697, −0.31711970818250304038173693924,
0.31711970818250304038173693924, 1.41637786386915938863599316697, 2.55792368017919531971202983591, 3.38408292042692404709436838387, 4.80237660831053476841172050769, 5.23369513811871983741929409031, 6.07413478674288249959382818130, 6.96960082658566039167644533487, 7.56432256108897651343660754090, 7.891775571617475677368519702224