| L(s) = 1 | − 2-s − 1.48·3-s + 4-s − 5-s + 1.48·6-s + 4.82·7-s − 8-s − 0.801·9-s + 10-s − 11-s − 1.48·12-s + 3.11·13-s − 4.82·14-s + 1.48·15-s + 16-s + 7.03·17-s + 0.801·18-s + 6.23·19-s − 20-s − 7.15·21-s + 22-s + 1.11·23-s + 1.48·24-s + 25-s − 3.11·26-s + 5.63·27-s + 4.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.856·3-s + 0.5·4-s − 0.447·5-s + 0.605·6-s + 1.82·7-s − 0.353·8-s − 0.267·9-s + 0.316·10-s − 0.301·11-s − 0.428·12-s + 0.864·13-s − 1.29·14-s + 0.382·15-s + 0.250·16-s + 1.70·17-s + 0.188·18-s + 1.42·19-s − 0.223·20-s − 1.56·21-s + 0.213·22-s + 0.232·23-s + 0.302·24-s + 0.200·25-s − 0.611·26-s + 1.08·27-s + 0.912·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.350437068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350437068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.48T + 3T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 7.03T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 - 1.11T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 5.76T + 41T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 9.27T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 9.80T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 0.459T + 97T^{2} \) |
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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
−8.297274044409084617300637977259, −7.63414549789270741169208505344, −7.14814983701671844481477708454, −5.92915294354694157983117209134, −5.43401820757890690891289298264, −4.87593322031272322420553615284, −3.74719316714301642095979354781, −2.78453290291344565872707480420, −1.42006492374519357981218225615, −0.863526906766281914854698705206,
0.863526906766281914854698705206, 1.42006492374519357981218225615, 2.78453290291344565872707480420, 3.74719316714301642095979354781, 4.87593322031272322420553615284, 5.43401820757890690891289298264, 5.92915294354694157983117209134, 7.14814983701671844481477708454, 7.63414549789270741169208505344, 8.297274044409084617300637977259
