L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 6·11-s − 5·13-s + 14-s + 16-s − 3·17-s − 6·22-s − 3·23-s − 5·25-s + 5·26-s − 28-s + 9·29-s + 4·31-s − 32-s + 3·34-s − 2·37-s + 8·43-s + 6·44-s + 3·46-s − 6·49-s + 5·50-s − 5·52-s − 3·53-s + 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.80·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.27·22-s − 0.625·23-s − 25-s + 0.980·26-s − 0.188·28-s + 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s + 1.21·43-s + 0.904·44-s + 0.442·46-s − 6/7·49-s + 0.707·50-s − 0.693·52-s − 0.412·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.72023666635394736173695745964, −6.85841529930894935996514770633, −6.52547725848872771498338713632, −5.77798205728261117317953598536, −4.61559374588543556536361618627, −4.07392909124551760489549274271, −3.01249692314631579373849129643, −2.19179284747142586381309074581, −1.22349109094330361666806860431, 0,
1.22349109094330361666806860431, 2.19179284747142586381309074581, 3.01249692314631579373849129643, 4.07392909124551760489549274271, 4.61559374588543556536361618627, 5.77798205728261117317953598536, 6.52547725848872771498338713632, 6.85841529930894935996514770633, 7.72023666635394736173695745964