| L(s) = 1 | − 1.82·2-s + 3-s + 1.33·4-s − 1.82·6-s + 1.44·7-s + 1.20·8-s + 9-s − 2.12·11-s + 1.33·12-s − 5.70·13-s − 2.64·14-s − 4.88·16-s + 4.15·17-s − 1.82·18-s + 1.70·19-s + 1.44·21-s + 3.89·22-s − 0.323·23-s + 1.20·24-s + 10.4·26-s + 27-s + 1.93·28-s − 8.74·29-s − 8.45·31-s + 6.50·32-s − 2.12·33-s − 7.59·34-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.669·4-s − 0.745·6-s + 0.546·7-s + 0.427·8-s + 0.333·9-s − 0.641·11-s + 0.386·12-s − 1.58·13-s − 0.705·14-s − 1.22·16-s + 1.00·17-s − 0.430·18-s + 0.390·19-s + 0.315·21-s + 0.829·22-s − 0.0674·23-s + 0.246·24-s + 2.04·26-s + 0.192·27-s + 0.365·28-s − 1.62·29-s − 1.51·31-s + 1.15·32-s − 0.370·33-s − 1.30·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 + 0.323T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 + 8.45T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 8.34T + 53T^{2} \) |
| 59 | \( 1 + 2.12T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 7.13T + 67T^{2} \) |
| 71 | \( 1 - 2.67T + 71T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 8.55T + 97T^{2} \) |
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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
−8.905767466265111573255116842723, −8.021482881025705419872424519872, −7.53203692575450239717073009276, −7.09106401508749150117503045913, −5.49587397555262134081745935286, −4.85362078670601144020044782698, −3.63432428239703359984648049004, −2.41596493930272092690263645912, −1.56517115121248049417204700700, 0,
1.56517115121248049417204700700, 2.41596493930272092690263645912, 3.63432428239703359984648049004, 4.85362078670601144020044782698, 5.49587397555262134081745935286, 7.09106401508749150117503045913, 7.53203692575450239717073009276, 8.021482881025705419872424519872, 8.905767466265111573255116842723
