| L(s) = 1 | − 2-s + 3.39·3-s + 4-s − 1.71·5-s − 3.39·6-s + 0.718·7-s − 8-s + 8.50·9-s + 1.71·10-s + 11-s + 3.39·12-s − 0.718·13-s − 0.718·14-s − 5.82·15-s + 16-s + 1.67·17-s − 8.50·18-s − 1.71·20-s + 2.43·21-s − 22-s + 1.11·23-s − 3.39·24-s − 2.04·25-s + 0.718·26-s + 18.6·27-s + 0.718·28-s + 8.17·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.95·3-s + 0.5·4-s − 0.768·5-s − 1.38·6-s + 0.271·7-s − 0.353·8-s + 2.83·9-s + 0.543·10-s + 0.301·11-s + 0.979·12-s − 0.199·13-s − 0.192·14-s − 1.50·15-s + 0.250·16-s + 0.405·17-s − 2.00·18-s − 0.384·20-s + 0.531·21-s − 0.213·22-s + 0.231·23-s − 0.692·24-s − 0.409·25-s + 0.140·26-s + 3.59·27-s + 0.135·28-s + 1.51·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.271344539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.271344539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 7 | \( 1 - 0.718T + 7T^{2} \) |
| 13 | \( 1 + 0.718T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 23 | \( 1 - 1.11T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 4.12T + 61T^{2} \) |
| 67 | \( 1 + 6.81T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 4.09T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−7.963429553250860042595713680787, −7.53036011825149460109680628439, −6.90084972989188823720327949947, −6.02011778073838328228656869973, −4.50177638210504712013245642410, −4.28182370979902261894868308081, −3.12831086390966373277756932688, −2.84080774109668872461600137458, −1.80118597259813148344029181559, −0.958263147803530099219741239877,
0.958263147803530099219741239877, 1.80118597259813148344029181559, 2.84080774109668872461600137458, 3.12831086390966373277756932688, 4.28182370979902261894868308081, 4.50177638210504712013245642410, 6.02011778073838328228656869973, 6.90084972989188823720327949947, 7.53036011825149460109680628439, 7.963429553250860042595713680787
