| L(s) = 1 | − 3-s − 4.88·7-s + 9-s + 6.11·11-s − 13-s + 2.88·17-s − 7.00·19-s + 4.88·21-s − 4.88·23-s − 27-s − 3.23·29-s + 9.77·31-s − 6.11·33-s + 9.89·37-s + 39-s − 9.89·41-s − 4·43-s + 5.77·47-s + 16.8·49-s − 2.88·51-s + 5.65·53-s + 7.00·57-s + 12.1·61-s − 4.88·63-s − 6.46·67-s + 4.88·69-s − 11.8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.84·7-s + 0.333·9-s + 1.84·11-s − 0.277·13-s + 0.700·17-s − 1.60·19-s + 1.06·21-s − 1.01·23-s − 0.192·27-s − 0.599·29-s + 1.75·31-s − 1.06·33-s + 1.62·37-s + 0.160·39-s − 1.54·41-s − 0.609·43-s + 0.842·47-s + 2.41·49-s − 0.404·51-s + 0.777·53-s + 0.928·57-s + 1.55·61-s − 0.615·63-s − 0.789·67-s + 0.588·69-s − 1.41·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4.88T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.88T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 5.11T + 97T^{2} \) |
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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
−7.19754059397209531726990316454, −6.65431305468119600928870934547, −6.18168776703387905654728020822, −5.78395954036553390677787135410, −4.44173500626481641236939759559, −3.99389288392653420459943815428, −3.25168627037502626658719082789, −2.25332611827170463168943344081, −1.06413965304453787467947673615, 0,
1.06413965304453787467947673615, 2.25332611827170463168943344081, 3.25168627037502626658719082789, 3.99389288392653420459943815428, 4.44173500626481641236939759559, 5.78395954036553390677787135410, 6.18168776703387905654728020822, 6.65431305468119600928870934547, 7.19754059397209531726990316454
