| L(s) = 1 | + 2-s + 3-s + 4-s + 2.32·5-s + 6-s − 0.374·7-s + 8-s + 9-s + 2.32·10-s + 5.67·11-s + 12-s − 13-s − 0.374·14-s + 2.32·15-s + 16-s − 6.54·17-s + 18-s + 5.75·19-s + 2.32·20-s − 0.374·21-s + 5.67·22-s + 4.54·23-s + 24-s + 0.400·25-s − 26-s + 27-s − 0.374·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.03·5-s + 0.408·6-s − 0.141·7-s + 0.353·8-s + 0.333·9-s + 0.734·10-s + 1.71·11-s + 0.288·12-s − 0.277·13-s − 0.100·14-s + 0.600·15-s + 0.250·16-s − 1.58·17-s + 0.235·18-s + 1.32·19-s + 0.519·20-s − 0.0817·21-s + 1.20·22-s + 0.948·23-s + 0.204·24-s + 0.0800·25-s − 0.196·26-s + 0.192·27-s − 0.0708·28-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
| \(L(1)\) |
\(\approx\) |
\(5.872524057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.872524057\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 + 0.374T + 7T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 + 7.19T + 31T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 1.27T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 5.86T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 + 5.01T + 73T^{2} \) |
| 79 | \( 1 - 3.90T + 79T^{2} \) |
| 83 | \( 1 + 2.10T + 83T^{2} \) |
| 89 | \( 1 + 3.18T + 89T^{2} \) |
| 97 | \( 1 + 4.91T + 97T^{2} \) |
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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
−7.64788134998235174559374581512, −6.86367887229582340490543388422, −6.54399062418132094122015104680, −5.76504057637805446880501612697, −4.97700853824281413948605370505, −4.25723071309659459857974539317, −3.52124381640036478048823678360, −2.70593513601915134036091471418, −1.93032536496956986854046634611, −1.14485984564208366572354886253,
1.14485984564208366572354886253, 1.93032536496956986854046634611, 2.70593513601915134036091471418, 3.52124381640036478048823678360, 4.25723071309659459857974539317, 4.97700853824281413948605370505, 5.76504057637805446880501612697, 6.54399062418132094122015104680, 6.86367887229582340490543388422, 7.64788134998235174559374581512
