| L(s) = 1 | + 2-s + 3-s + 4-s − 3.68·5-s + 6-s + 3.83·7-s + 8-s + 9-s − 3.68·10-s − 11-s + 12-s − 4.35·13-s + 3.83·14-s − 3.68·15-s + 16-s + 5.53·17-s + 18-s − 2.05·19-s − 3.68·20-s + 3.83·21-s − 22-s − 4.26·23-s + 24-s + 8.56·25-s − 4.35·26-s + 27-s + 3.83·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.64·5-s + 0.408·6-s + 1.44·7-s + 0.353·8-s + 0.333·9-s − 1.16·10-s − 0.301·11-s + 0.288·12-s − 1.20·13-s + 1.02·14-s − 0.950·15-s + 0.250·16-s + 1.34·17-s + 0.235·18-s − 0.470·19-s − 0.823·20-s + 0.835·21-s − 0.213·22-s − 0.890·23-s + 0.204·24-s + 1.71·25-s − 0.854·26-s + 0.192·27-s + 0.723·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.226351064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.226351064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 + 4.26T + 23T^{2} \) |
| 29 | \( 1 - 3.03T + 29T^{2} \) |
| 31 | \( 1 - 9.95T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 3.94T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 67 | \( 1 + 5.14T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 + 2.58T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 - 7.97T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.997480107510176255362127970209, −7.81611262223346348218132458868, −7.35860305387636926205502158788, −6.20702622848150926524354212404, −5.09026513739807643382205928637, −4.53784785098150253360957514528, −4.04067337783164647614488843044, −3.04061676266762195898243303913, −2.26971931017383605098158062265, −0.921836606494412742066908515995,
0.921836606494412742066908515995, 2.26971931017383605098158062265, 3.04061676266762195898243303913, 4.04067337783164647614488843044, 4.53784785098150253360957514528, 5.09026513739807643382205928637, 6.20702622848150926524354212404, 7.35860305387636926205502158788, 7.81611262223346348218132458868, 7.997480107510176255362127970209
