| L(s) = 1 | − 2-s − 3-s + 4-s + 0.448·5-s + 6-s + 1.73·7-s − 8-s + 9-s − 0.448·10-s + 1.41·11-s − 12-s − 13-s − 1.73·14-s − 0.448·15-s + 16-s − 1.53·17-s − 18-s + 6.26·19-s + 0.448·20-s − 1.73·21-s − 1.41·22-s + 4.01·23-s + 24-s − 4.79·25-s + 26-s − 27-s + 1.73·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.200·5-s + 0.408·6-s + 0.657·7-s − 0.353·8-s + 0.333·9-s − 0.141·10-s + 0.425·11-s − 0.288·12-s − 0.277·13-s − 0.465·14-s − 0.115·15-s + 0.250·16-s − 0.371·17-s − 0.235·18-s + 1.43·19-s + 0.100·20-s − 0.379·21-s − 0.300·22-s + 0.837·23-s + 0.204·24-s − 0.959·25-s + 0.196·26-s − 0.192·27-s + 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 0.448T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 23 | \( 1 - 4.01T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 - 0.153T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 - 7.24T + 53T^{2} \) |
| 59 | \( 1 + 2.88T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 5.32T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 0.364T + 97T^{2} \) |
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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
−7.38176895571486684057950355360, −7.00448771808504135231670933820, −6.16438641800115900196546719942, −5.37906297198911304232617149238, −4.92118830589820023200151500950, −3.87382570923885206348089424563, −3.01329161671675494903395554036, −1.86134595869135273821497558282, −1.26842616154523329745994630319, 0,
1.26842616154523329745994630319, 1.86134595869135273821497558282, 3.01329161671675494903395554036, 3.87382570923885206348089424563, 4.92118830589820023200151500950, 5.37906297198911304232617149238, 6.16438641800115900196546719942, 7.00448771808504135231670933820, 7.38176895571486684057950355360
