| L(s) = 1 | − 2-s + 4-s + 0.350·5-s + 3.21·7-s − 8-s − 0.350·10-s + 3.33·11-s − 4.79·13-s − 3.21·14-s + 16-s − 1.65·17-s − 6.44·19-s + 0.350·20-s − 3.33·22-s + 7.13·23-s − 4.87·25-s + 4.79·26-s + 3.21·28-s + 2.80·29-s + 6.66·31-s − 32-s + 1.65·34-s + 1.12·35-s − 0.669·37-s + 6.44·38-s − 0.350·40-s + 0.589·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.156·5-s + 1.21·7-s − 0.353·8-s − 0.110·10-s + 1.00·11-s − 1.32·13-s − 0.859·14-s + 0.250·16-s − 0.402·17-s − 1.47·19-s + 0.0783·20-s − 0.710·22-s + 1.48·23-s − 0.975·25-s + 0.940·26-s + 0.607·28-s + 0.521·29-s + 1.19·31-s − 0.176·32-s + 0.284·34-s + 0.190·35-s − 0.110·37-s + 1.04·38-s − 0.0553·40-s + 0.0921·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.590667067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590667067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 0.350T + 5T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 - 7.13T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + 0.669T + 37T^{2} \) |
| 41 | \( 1 - 0.589T + 41T^{2} \) |
| 43 | \( 1 + 4.08T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 - 9.99T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 6.63T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 - 0.107T + 89T^{2} \) |
| 97 | \( 1 + 0.0990T + 97T^{2} \) |
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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
−8.350213327214360784687400251572, −7.967097574028092227570637320382, −6.88884321602462086551025085847, −6.61884372361050168544782671737, −5.41952886941451031886193080039, −4.70602374058370728210786157169, −3.95456256681993756530944113265, −2.55339771688022433760893281687, −1.93642166299186136932780364387, −0.817260842173802649091310137208,
0.817260842173802649091310137208, 1.93642166299186136932780364387, 2.55339771688022433760893281687, 3.95456256681993756530944113265, 4.70602374058370728210786157169, 5.41952886941451031886193080039, 6.61884372361050168544782671737, 6.88884321602462086551025085847, 7.967097574028092227570637320382, 8.350213327214360784687400251572
