| L(s) = 1 | + 0.773·5-s − 1.91·7-s + 6.46·11-s + 0.0817·13-s − 3·17-s + 19-s − 0.773·23-s − 4.40·25-s − 3.77·29-s + 0.918·31-s − 1.48·35-s − 9.23·37-s − 2.93·41-s − 6.32·43-s + 4.54·47-s − 3.32·49-s − 3.30·53-s + 5·55-s − 3.40·59-s + 0.0817·61-s + 0.0631·65-s − 6.40·67-s + 4.79·71-s + 11.4·73-s − 12.4·77-s − 9.23·79-s + 3·83-s + ⋯ |
| L(s) = 1 | + 0.345·5-s − 0.725·7-s + 1.94·11-s + 0.0226·13-s − 0.727·17-s + 0.229·19-s − 0.161·23-s − 0.880·25-s − 0.700·29-s + 0.164·31-s − 0.250·35-s − 1.51·37-s − 0.458·41-s − 0.963·43-s + 0.663·47-s − 0.474·49-s − 0.454·53-s + 0.674·55-s − 0.442·59-s + 0.0104·61-s + 0.00783·65-s − 0.782·67-s + 0.568·71-s + 1.34·73-s − 1.41·77-s − 1.03·79-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 0.773T + 5T^{2} \) |
| 7 | \( 1 + 1.91T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 - 0.0817T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 + 0.773T + 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 - 0.918T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 + 2.93T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 - 0.0817T + 61T^{2} \) |
| 67 | \( 1 + 6.40T + 67T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 9.23T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 - 1.40T + 97T^{2} \) |
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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
−7.24231407776750425742959647844, −6.70592158084894562821049613698, −6.23272316259352024251324069134, −5.51598519685138981506848327872, −4.55293391017196284967651911357, −3.79043544061436742762258061340, −3.28477146389995705557207738303, −2.07453838364489735172406010877, −1.37964781875537455540209408672, 0,
1.37964781875537455540209408672, 2.07453838364489735172406010877, 3.28477146389995705557207738303, 3.79043544061436742762258061340, 4.55293391017196284967651911357, 5.51598519685138981506848327872, 6.23272316259352024251324069134, 6.70592158084894562821049613698, 7.24231407776750425742959647844
