| L(s) = 1 | − 2.41·2-s + 3-s + 3.82·4-s − 2.41·6-s − 4.41·8-s + 9-s + 2.82·11-s + 3.82·12-s − 4.82·13-s + 2.99·16-s − 7.65·17-s − 2.41·18-s + 0.828·19-s − 6.82·22-s + 7.65·23-s − 4.41·24-s + 11.6·26-s + 27-s + 6·29-s − 6.48·31-s + 1.58·32-s + 2.82·33-s + 18.4·34-s + 3.82·36-s + 3.65·37-s − 1.99·38-s − 4.82·39-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.91·4-s − 0.985·6-s − 1.56·8-s + 0.333·9-s + 0.852·11-s + 1.10·12-s − 1.33·13-s + 0.749·16-s − 1.85·17-s − 0.569·18-s + 0.190·19-s − 1.45·22-s + 1.59·23-s − 0.901·24-s + 2.28·26-s + 0.192·27-s + 1.11·29-s − 1.16·31-s + 0.280·32-s + 0.492·33-s + 3.17·34-s + 0.638·36-s + 0.601·37-s − 0.324·38-s − 0.773·39-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 | \( 1 + 2.41T + 2T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−8.470846410611901895510491657487, −7.49544932829422530169818677838, −6.92810869455272734225464159997, −6.57105880503012754295085468260, −5.10989189898427978899771339043, −4.26632518553639901508329464596, −2.98054253741822814168836985063, −2.24522908435688043043129505352, −1.33174646643943670575706273319, 0,
1.33174646643943670575706273319, 2.24522908435688043043129505352, 2.98054253741822814168836985063, 4.26632518553639901508329464596, 5.10989189898427978899771339043, 6.57105880503012754295085468260, 6.92810869455272734225464159997, 7.49544932829422530169818677838, 8.470846410611901895510491657487
