| L(s) = 1 | − 2.10·2-s − 1.77·3-s + 2.44·4-s − 5-s + 3.75·6-s + 3.19·7-s − 0.946·8-s + 0.162·9-s + 2.10·10-s − 0.544·11-s − 4.35·12-s − 3.74·13-s − 6.74·14-s + 1.77·15-s − 2.90·16-s + 4.20·17-s − 0.343·18-s − 5.83·19-s − 2.44·20-s − 5.68·21-s + 1.14·22-s + 3.25·23-s + 1.68·24-s + 25-s + 7.90·26-s + 5.04·27-s + 7.82·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s − 1.02·3-s + 1.22·4-s − 0.447·5-s + 1.53·6-s + 1.20·7-s − 0.334·8-s + 0.0542·9-s + 0.666·10-s − 0.164·11-s − 1.25·12-s − 1.03·13-s − 1.80·14-s + 0.459·15-s − 0.725·16-s + 1.02·17-s − 0.0808·18-s − 1.33·19-s − 0.547·20-s − 1.24·21-s + 0.244·22-s + 0.677·23-s + 0.343·24-s + 0.200·25-s + 1.55·26-s + 0.971·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 + 0.544T + 11T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 - 0.823T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 5.90T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 + 6.94T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 + 9.66T + 83T^{2} \) |
| 89 | \( 1 - 2.06T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
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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
−9.258854718185019048168846556967, −8.532229947712798966521644853011, −7.71121273534814212721997277090, −7.27058319045587758586808588590, −6.11343266148051980092617078880, −5.11030033234971505413684503092, −4.37841371344682712353140860864, −2.53288714755462651364640504043, −1.24559113431466233165579139074, 0,
1.24559113431466233165579139074, 2.53288714755462651364640504043, 4.37841371344682712353140860864, 5.11030033234971505413684503092, 6.11343266148051980092617078880, 7.27058319045587758586808588590, 7.71121273534814212721997277090, 8.532229947712798966521644853011, 9.258854718185019048168846556967
