| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 8.30·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s + 16.6·10-s − 14.5·11-s + 12·12-s − 22.1·13-s + 14·14-s − 24.9·15-s + 16·16-s − 72.5·17-s − 18·18-s + 97.6·19-s − 33.2·20-s − 21·21-s + 29.0·22-s + 23·23-s − 24·24-s − 55.9·25-s + 44.3·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.743·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.525·10-s − 0.397·11-s + 0.288·12-s − 0.472·13-s + 0.267·14-s − 0.429·15-s + 0.250·16-s − 1.03·17-s − 0.235·18-s + 1.17·19-s − 0.371·20-s − 0.218·21-s + 0.281·22-s + 0.208·23-s − 0.204·24-s − 0.447·25-s + 0.334·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.108341997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108341997\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 8.30T + 125T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 269.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 155.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 92.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 347.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 391.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 271.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 230.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 766.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 986.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 566.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 153.T + 9.12e5T^{2} \) |
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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
−9.503149105196015850832183445897, −8.861916074291757073314006720205, −7.904282391909392643324817272915, −7.41652010317643074535040015354, −6.54008134916776142851004587033, −5.28638682210765606203483893402, −4.09577956158972322814933655967, −3.11447698521634877549845780952, −2.12119925861173689591365627248, −0.59023247568240138206342734373,
0.59023247568240138206342734373, 2.12119925861173689591365627248, 3.11447698521634877549845780952, 4.09577956158972322814933655967, 5.28638682210765606203483893402, 6.54008134916776142851004587033, 7.41652010317643074535040015354, 7.904282391909392643324817272915, 8.861916074291757073314006720205, 9.503149105196015850832183445897
