L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 6·11-s − 13-s + 14-s + 16-s − 3·17-s − 4·19-s − 6·22-s + 3·23-s − 26-s + 28-s − 3·29-s + 5·31-s + 32-s − 3·34-s − 10·37-s − 4·38-s − 9·41-s − 43-s − 6·44-s + 3·46-s + 49-s − 52-s − 9·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.80·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 1.27·22-s + 0.625·23-s − 0.196·26-s + 0.188·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.514·34-s − 1.64·37-s − 0.648·38-s − 1.40·41-s − 0.152·43-s − 0.904·44-s + 0.442·46-s + 1/7·49-s − 0.138·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.254055372016988552354997842377, −7.47916616075727475083077091562, −6.78411226586307184417191810376, −5.90989378861191419538786855155, −4.98754538022209224421783767230, −4.72462107827192886206006532713, −3.50396926841669060205873743357, −2.62679874406622467537197418542, −1.83210176479960116711288727534, 0,
1.83210176479960116711288727534, 2.62679874406622467537197418542, 3.50396926841669060205873743357, 4.72462107827192886206006532713, 4.98754538022209224421783767230, 5.90989378861191419538786855155, 6.78411226586307184417191810376, 7.47916616075727475083077091562, 8.254055372016988552354997842377