L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 2·13-s + 14-s + 16-s − 6·17-s − 4·19-s − 20-s + 22-s + 25-s + 2·26-s + 28-s − 6·29-s − 4·31-s + 32-s − 6·34-s − 35-s + 2·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.48329804208154492790279469993, −6.78341449865007631667304257093, −6.25056292819033940130397386385, −5.41023435376890073002985541022, −4.61268938691383493592448675430, −4.05440746228886283487670904194, −3.37670554394465108926490957156, −2.29463461204304627579368908329, −1.55204340038518250112867700620, 0,
1.55204340038518250112867700620, 2.29463461204304627579368908329, 3.37670554394465108926490957156, 4.05440746228886283487670904194, 4.61268938691383493592448675430, 5.41023435376890073002985541022, 6.25056292819033940130397386385, 6.78341449865007631667304257093, 7.48329804208154492790279469993