| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s − 11-s − 12-s − 13-s − 4·14-s + 16-s + 18-s + 4·21-s − 22-s − 24-s − 5·25-s − 26-s − 27-s − 4·28-s + 10·31-s + 32-s + 33-s + 36-s − 2·37-s + 39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.235·18-s + 0.872·21-s − 0.213·22-s − 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + 1.79·31-s + 0.176·32-s + 0.174·33-s + 1/6·36-s − 0.328·37-s + 0.160·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.89570737138457, −12.42441090119934, −12.10996829997264, −11.51642928425832, −11.33683332434043, −10.47857174998710, −10.15220794108860, −9.921047763361972, −9.364902588549857, −8.807262244254081, −8.128285634799855, −7.704218244604616, −7.050699798248931, −6.742926455709621, −6.215454316255139, −5.888748383532460, −5.440814626645357, −4.737582377171633, −4.366079025837795, −3.816165185964158, −3.041558308394875, −2.985914510922837, −2.157792240584002, −1.497437123217246, −0.6322373790021966, 0,
0.6322373790021966, 1.497437123217246, 2.157792240584002, 2.985914510922837, 3.041558308394875, 3.816165185964158, 4.366079025837795, 4.737582377171633, 5.440814626645357, 5.888748383532460, 6.215454316255139, 6.742926455709621, 7.050699798248931, 7.704218244604616, 8.128285634799855, 8.807262244254081, 9.364902588549857, 9.921047763361972, 10.15220794108860, 10.47857174998710, 11.33683332434043, 11.51642928425832, 12.10996829997264, 12.42441090119934, 12.89570737138457
