| L(s) = 1 | + 2-s − 0.490·3-s + 4-s − 5-s − 0.490·6-s − 1.29·7-s + 8-s − 2.75·9-s − 10-s − 11-s − 0.490·12-s + 6.36·13-s − 1.29·14-s + 0.490·15-s + 16-s − 5.15·17-s − 2.75·18-s − 0.102·19-s − 20-s + 0.634·21-s − 22-s − 0.876·23-s − 0.490·24-s + 25-s + 6.36·26-s + 2.82·27-s − 1.29·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.283·3-s + 0.5·4-s − 0.447·5-s − 0.200·6-s − 0.489·7-s + 0.353·8-s − 0.919·9-s − 0.316·10-s − 0.301·11-s − 0.141·12-s + 1.76·13-s − 0.345·14-s + 0.126·15-s + 0.250·16-s − 1.25·17-s − 0.650·18-s − 0.0234·19-s − 0.223·20-s + 0.138·21-s − 0.213·22-s − 0.182·23-s − 0.100·24-s + 0.200·25-s + 1.24·26-s + 0.543·27-s − 0.244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068652281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068652281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 0.490T + 3T^{2} \) |
| 7 | \( 1 + 1.29T + 7T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 + 0.102T + 19T^{2} \) |
| 23 | \( 1 + 0.876T + 23T^{2} \) |
| 29 | \( 1 - 7.52T + 29T^{2} \) |
| 31 | \( 1 - 8.76T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 3.60T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 + 0.175T + 53T^{2} \) |
| 59 | \( 1 + 5.90T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 79 | \( 1 - 3.97T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
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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
−7.897349172243893365571608994826, −6.78758216295597285686347225472, −6.30102167334537401818873099629, −5.97447515534216945298688224935, −4.84308656887936849790850356403, −4.46073999008007534568508081401, −3.31402347999382116604984948490, −3.08446994710320296719145908440, −1.89514224732188029434242168351, −0.63358656756114165425014042830,
0.63358656756114165425014042830, 1.89514224732188029434242168351, 3.08446994710320296719145908440, 3.31402347999382116604984948490, 4.46073999008007534568508081401, 4.84308656887936849790850356403, 5.97447515534216945298688224935, 6.30102167334537401818873099629, 6.78758216295597285686347225472, 7.897349172243893365571608994826
