L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 5·7-s + 8-s + 9-s + 11-s + 12-s − 5·14-s + 16-s − 3·17-s + 18-s + 8·19-s − 5·21-s + 22-s − 8·23-s + 24-s + 27-s − 5·28-s − 5·29-s + 3·31-s + 32-s + 33-s − 3·34-s + 36-s + 37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.83·19-s − 1.09·21-s + 0.213·22-s − 1.66·23-s + 0.204·24-s + 0.192·27-s − 0.944·28-s − 0.928·29-s + 0.538·31-s + 0.176·32-s + 0.174·33-s − 0.514·34-s + 1/6·36-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−7.61126580863542575635341746296, −6.96529118440617002689426008676, −6.33964692250627541440653674144, −5.77032413368424293663574141289, −4.78593868575399903884036652436, −3.80232955621652237767760797034, −3.36642414199451163299052866034, −2.69031361493034312998525267374, −1.60469718942117829933875115367, 0,
1.60469718942117829933875115367, 2.69031361493034312998525267374, 3.36642414199451163299052866034, 3.80232955621652237767760797034, 4.78593868575399903884036652436, 5.77032413368424293663574141289, 6.33964692250627541440653674144, 6.96529118440617002689426008676, 7.61126580863542575635341746296