| L(s) = 1 | − 0.823·2-s − 3-s − 1.32·4-s − 5-s + 0.823·6-s + 2.12·7-s + 2.73·8-s + 9-s + 0.823·10-s + 5.61·11-s + 1.32·12-s − 4.36·13-s − 1.75·14-s + 15-s + 0.394·16-s − 0.824·17-s − 0.823·18-s − 2.24·19-s + 1.32·20-s − 2.12·21-s − 4.62·22-s + 5.33·23-s − 2.73·24-s + 25-s + 3.59·26-s − 27-s − 2.81·28-s + ⋯ |
| L(s) = 1 | − 0.582·2-s − 0.577·3-s − 0.661·4-s − 0.447·5-s + 0.336·6-s + 0.803·7-s + 0.966·8-s + 0.333·9-s + 0.260·10-s + 1.69·11-s + 0.381·12-s − 1.21·13-s − 0.467·14-s + 0.258·15-s + 0.0985·16-s − 0.199·17-s − 0.194·18-s − 0.515·19-s + 0.295·20-s − 0.464·21-s − 0.986·22-s + 1.11·23-s − 0.558·24-s + 0.200·25-s + 0.705·26-s − 0.192·27-s − 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.823T + 2T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 + 0.824T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + 2.66T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 + 1.57T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 - 0.220T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 4.83T + 97T^{2} \) |
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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
−7.76539799839597337139168780156, −7.06639336389262669176626153114, −6.55371415044525628023036315781, −5.36263088904418371208963211309, −4.78655600355388563775037432461, −4.25669947014301476753222481069, −3.42669571985813254385211949466, −1.91838216097483249695902035763, −1.13287579940339486009632624960, 0,
1.13287579940339486009632624960, 1.91838216097483249695902035763, 3.42669571985813254385211949466, 4.25669947014301476753222481069, 4.78655600355388563775037432461, 5.36263088904418371208963211309, 6.55371415044525628023036315781, 7.06639336389262669176626153114, 7.76539799839597337139168780156
