| L(s) = 1 | + 1.13·2-s − 0.718·4-s + 4.11·7-s − 3.07·8-s + 5.78·11-s − 6.78·13-s + 4.65·14-s − 2.04·16-s − 5.41·17-s − 19-s + 6.54·22-s − 8.04·23-s − 7.67·26-s − 2.95·28-s − 5.34·29-s + 0.327·31-s + 3.83·32-s − 6.12·34-s − 10.7·37-s − 1.13·38-s + 2.70·41-s − 0.654·43-s − 4.15·44-s − 9.11·46-s + 7.67·47-s + 9.89·49-s + 4.87·52-s + ⋯ |
| L(s) = 1 | + 0.800·2-s − 0.359·4-s + 1.55·7-s − 1.08·8-s + 1.74·11-s − 1.88·13-s + 1.24·14-s − 0.511·16-s − 1.31·17-s − 0.229·19-s + 1.39·22-s − 1.67·23-s − 1.50·26-s − 0.558·28-s − 0.991·29-s + 0.0587·31-s + 0.678·32-s − 1.05·34-s − 1.77·37-s − 0.183·38-s + 0.422·41-s − 0.0998·43-s − 0.626·44-s − 1.34·46-s + 1.11·47-s + 1.41·49-s + 0.676·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 7 | \( 1 - 4.11T + 7T^{2} \) |
| 11 | \( 1 - 5.78T + 11T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 - 0.327T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 0.654T + 43T^{2} \) |
| 47 | \( 1 - 7.67T + 47T^{2} \) |
| 53 | \( 1 - 0.813T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 + 7.87T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.01T + 79T^{2} \) |
| 83 | \( 1 - 1.25T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 6.78T + 97T^{2} \) |
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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
−8.006598229774360203817862177516, −7.23922689999852654070941443038, −6.45792918194162190556727320396, −5.59830079834284482638901528374, −4.84543554291220786294898285362, −4.31982481393121260146985680627, −3.79504226843718266159077802331, −2.38680933613079856995051056595, −1.67768522572745865857345139820, 0,
1.67768522572745865857345139820, 2.38680933613079856995051056595, 3.79504226843718266159077802331, 4.31982481393121260146985680627, 4.84543554291220786294898285362, 5.59830079834284482638901528374, 6.45792918194162190556727320396, 7.23922689999852654070941443038, 8.006598229774360203817862177516
