| L(s) = 1 | − 2.27·2-s − 3-s + 3.15·4-s + 2.27·6-s − 2.63·8-s + 9-s + 5.05·11-s − 3.15·12-s − 0.224·13-s − 0.338·16-s + 3.59·17-s − 2.27·18-s + 4.78·19-s − 11.4·22-s − 7.01·23-s + 2.63·24-s + 0.510·26-s − 27-s − 8.74·29-s − 8.78·31-s + 6.03·32-s − 5.05·33-s − 8.16·34-s + 3.15·36-s + 5.91·37-s − 10.8·38-s + 0.224·39-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 0.577·3-s + 1.57·4-s + 0.927·6-s − 0.930·8-s + 0.333·9-s + 1.52·11-s − 0.911·12-s − 0.0623·13-s − 0.0847·16-s + 0.872·17-s − 0.535·18-s + 1.09·19-s − 2.44·22-s − 1.46·23-s + 0.537·24-s + 0.100·26-s − 0.192·27-s − 1.62·29-s − 1.57·31-s + 1.06·32-s − 0.879·33-s − 1.40·34-s + 0.526·36-s + 0.972·37-s − 1.76·38-s + 0.0359·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 + 0.224T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 + 7.01T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 + 8.78T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 - 4.31T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 0.361T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 - 8.36T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 4.86T + 73T^{2} \) |
| 79 | \( 1 - 2.88T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−8.190487035951524731853001891456, −7.48707497102337863330752275881, −6.98717965964678320282202837710, −6.10636266466738178224636452986, −5.46186041313615928355416106259, −4.20824535472512655445647878011, −3.38269209353119355880330228715, −1.86605648499940987029009787770, −1.26457750785897733229748511893, 0,
1.26457750785897733229748511893, 1.86605648499940987029009787770, 3.38269209353119355880330228715, 4.20824535472512655445647878011, 5.46186041313615928355416106259, 6.10636266466738178224636452986, 6.98717965964678320282202837710, 7.48707497102337863330752275881, 8.190487035951524731853001891456
