| L(s) = 1 | − 0.0240·2-s − 2.93·3-s − 1.99·4-s + 0.0704·6-s + 1.66·7-s + 0.0960·8-s + 5.58·9-s + 3.33·11-s + 5.85·12-s − 0.0400·14-s + 3.99·16-s − 7.07·17-s − 0.134·18-s − 6.68·19-s − 4.87·21-s − 0.0801·22-s − 4.02·23-s − 0.281·24-s − 7.57·27-s − 3.32·28-s − 0.000401·29-s + 4.14·31-s − 0.288·32-s − 9.76·33-s + 0.170·34-s − 11.1·36-s + 8.96·37-s + ⋯ |
| L(s) = 1 | − 0.0169·2-s − 1.69·3-s − 0.999·4-s + 0.0287·6-s + 0.629·7-s + 0.0339·8-s + 1.86·9-s + 1.00·11-s + 1.69·12-s − 0.0106·14-s + 0.999·16-s − 1.71·17-s − 0.0316·18-s − 1.53·19-s − 1.06·21-s − 0.0170·22-s − 0.840·23-s − 0.0574·24-s − 1.45·27-s − 0.629·28-s − 7.44e − 5·29-s + 0.744·31-s − 0.0509·32-s − 1.70·33-s + 0.0291·34-s − 1.86·36-s + 1.47·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.0240T + 2T^{2} \) |
| 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 + 6.68T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 + 0.000401T + 29T^{2} \) |
| 31 | \( 1 - 4.14T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 + 0.958T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 - 7.99T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 1.97T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 + 3.64T + 83T^{2} \) |
| 89 | \( 1 + 0.989T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.130998164441674288491676890399, −7.10615672618754437348175333513, −6.19111751520067119446148781485, −6.05907823107840902721058034021, −4.84099725584611440046891272251, −4.45359302167716538162971309496, −3.98220831027949613568709608525, −2.15044411801662563417943704705, −1.01934546006553507399045938549, 0,
1.01934546006553507399045938549, 2.15044411801662563417943704705, 3.98220831027949613568709608525, 4.45359302167716538162971309496, 4.84099725584611440046891272251, 6.05907823107840902721058034021, 6.19111751520067119446148781485, 7.10615672618754437348175333513, 8.130998164441674288491676890399
