| L(s) = 1 | + 2-s + 4-s + 3.23·7-s + 8-s − 4.47·11-s + 13-s + 3.23·14-s + 16-s − 7.23·17-s − 2.76·19-s − 4.47·22-s − 2.76·23-s + 26-s + 3.23·28-s + 3.70·29-s − 4·31-s + 32-s − 7.23·34-s − 10.9·37-s − 2.76·38-s − 3.52·41-s − 2.47·43-s − 4.47·44-s − 2.76·46-s − 12.9·47-s + 3.47·49-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.22·7-s + 0.353·8-s − 1.34·11-s + 0.277·13-s + 0.864·14-s + 0.250·16-s − 1.75·17-s − 0.634·19-s − 0.953·22-s − 0.576·23-s + 0.196·26-s + 0.611·28-s + 0.688·29-s − 0.718·31-s + 0.176·32-s − 1.24·34-s − 1.79·37-s − 0.448·38-s − 0.550·41-s − 0.376·43-s − 0.674·44-s − 0.407·46-s − 1.88·47-s + 0.496·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 0.472T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−7.81122735492602141585430979018, −6.90307643933691920894558790339, −6.34191398845210435619100740480, −5.29071582008442050734130698629, −4.93033789994839879741999634505, −4.24529391010141017319634547131, −3.29715971240276180006829501449, −2.23689570976657108643725636678, −1.75756537985335038055929929512, 0,
1.75756537985335038055929929512, 2.23689570976657108643725636678, 3.29715971240276180006829501449, 4.24529391010141017319634547131, 4.93033789994839879741999634505, 5.29071582008442050734130698629, 6.34191398845210435619100740480, 6.90307643933691920894558790339, 7.81122735492602141585430979018
