| L(s) = 1 | + 2.42·2-s − 3-s + 3.87·4-s + 5-s − 2.42·6-s + 4.53·8-s + 9-s + 2.42·10-s − 11-s − 3.87·12-s − 6.70·13-s − 15-s + 3.24·16-s − 5.74·17-s + 2.42·18-s + 1.83·19-s + 3.87·20-s − 2.42·22-s + 7.33·23-s − 4.53·24-s + 25-s − 16.2·26-s − 27-s − 6.84·29-s − 2.42·30-s − 0.852·31-s − 1.21·32-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.577·3-s + 1.93·4-s + 0.447·5-s − 0.989·6-s + 1.60·8-s + 0.333·9-s + 0.766·10-s − 0.301·11-s − 1.11·12-s − 1.85·13-s − 0.258·15-s + 0.810·16-s − 1.39·17-s + 0.571·18-s + 0.422·19-s + 0.865·20-s − 0.516·22-s + 1.52·23-s − 0.925·24-s + 0.200·25-s − 3.18·26-s − 0.192·27-s − 1.27·29-s − 0.442·30-s − 0.153·31-s − 0.214·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 + 6.84T + 29T^{2} \) |
| 31 | \( 1 + 0.852T + 31T^{2} \) |
| 37 | \( 1 + 4.17T + 37T^{2} \) |
| 41 | \( 1 - 0.963T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 0.892T + 71T^{2} \) |
| 73 | \( 1 + 16.7T + 73T^{2} \) |
| 79 | \( 1 + 0.729T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 3.70T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.14105698017991575232700330271, −6.67691270465215480559769885620, −5.79011498133138028759051142023, −5.31163840756353874478137218879, −4.72291734608468666208301481406, −4.24850494875139372649658332232, −3.09683991314522185413939748337, −2.52461915802333247091974256599, −1.70142670709045762509885758487, 0,
1.70142670709045762509885758487, 2.52461915802333247091974256599, 3.09683991314522185413939748337, 4.24850494875139372649658332232, 4.72291734608468666208301481406, 5.31163840756353874478137218879, 5.79011498133138028759051142023, 6.67691270465215480559769885620, 7.14105698017991575232700330271
