| L(s) = 1 | − 2-s − 4-s − 5-s + 3·7-s + 3·8-s + 10-s − 4·13-s − 3·14-s − 16-s − 4·19-s + 20-s + 8·23-s + 25-s + 4·26-s − 3·28-s + 6·29-s − 2·31-s − 5·32-s − 3·35-s − 8·37-s + 4·38-s − 3·40-s − 5·41-s − 5·43-s − 8·46-s + 3·47-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.13·7-s + 1.06·8-s + 0.316·10-s − 1.10·13-s − 0.801·14-s − 1/4·16-s − 0.917·19-s + 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.784·26-s − 0.566·28-s + 1.11·29-s − 0.359·31-s − 0.883·32-s − 0.507·35-s − 1.31·37-s + 0.648·38-s − 0.474·40-s − 0.780·41-s − 0.762·43-s − 1.17·46-s + 0.437·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.968139580675606583215130365666, −7.26031167725113291219456161735, −6.73793036347671686248541698351, −5.32660623389401820520516813497, −4.86010043841174802178759439385, −4.33642482285914585917613060074, −3.25561062576247856022902752208, −2.11090774412536995750139563450, −1.17894778875075887378558675851, 0,
1.17894778875075887378558675851, 2.11090774412536995750139563450, 3.25561062576247856022902752208, 4.33642482285914585917613060074, 4.86010043841174802178759439385, 5.32660623389401820520516813497, 6.73793036347671686248541698351, 7.26031167725113291219456161735, 7.968139580675606583215130365666
