| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 3·13-s + 16-s − 3·17-s − 18-s − 2·19-s + 3·23-s + 24-s + 3·26-s − 27-s + 7·29-s + 31-s − 32-s + 3·34-s + 36-s + 4·37-s + 2·38-s + 3·39-s − 5·41-s − 3·43-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.625·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s + 1.29·29-s + 0.179·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s + 0.657·37-s + 0.324·38-s + 0.480·39-s − 0.780·41-s − 0.457·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.50719427347713378186589387454, −6.79605216876614117154123731272, −6.46562506061061629770262682243, −5.45650110232807130353463171851, −4.82562261306925169632843566683, −4.04146108607793005359969991558, −2.89623472978449929062216728404, −2.16318433373139988479669974841, −1.06307838916259554679920503600, 0,
1.06307838916259554679920503600, 2.16318433373139988479669974841, 2.89623472978449929062216728404, 4.04146108607793005359969991558, 4.82562261306925169632843566683, 5.45650110232807130353463171851, 6.46562506061061629770262682243, 6.79605216876614117154123731272, 7.50719427347713378186589387454
