L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2·7-s − 3·8-s + 9-s + 12-s − 2·13-s + 2·14-s − 16-s + 18-s − 2·19-s − 2·21-s + 23-s + 3·24-s − 5·25-s − 2·26-s − 27-s − 2·28-s + 10·29-s + 4·31-s + 5·32-s − 36-s + 2·37-s − 2·38-s + 2·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.436·21-s + 0.208·23-s + 0.612·24-s − 25-s − 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.85·29-s + 0.718·31-s + 0.883·32-s − 1/6·36-s + 0.328·37-s − 0.324·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.43700395959370847730814508110, −6.40518653025417471032971165541, −6.10879390051279695888981242477, −5.03280439513537250484181575419, −4.82513226995920549143756999435, −4.16237501407393666057325868935, −3.23738062413275121702544540117, −2.33395980230393732187994680066, −1.17743591893625891330823714552, 0,
1.17743591893625891330823714552, 2.33395980230393732187994680066, 3.23738062413275121702544540117, 4.16237501407393666057325868935, 4.82513226995920549143756999435, 5.03280439513537250484181575419, 6.10879390051279695888981242477, 6.40518653025417471032971165541, 7.43700395959370847730814508110