| L(s) = 1 | − 0.577·2-s − 0.746·3-s − 1.66·4-s + 5-s + 0.431·6-s − 4.19·7-s + 2.11·8-s − 2.44·9-s − 0.577·10-s − 11-s + 1.24·12-s + 0.324·13-s + 2.42·14-s − 0.746·15-s + 2.10·16-s − 5.00·17-s + 1.41·18-s − 19-s − 1.66·20-s + 3.13·21-s + 0.577·22-s + 8.11·23-s − 1.58·24-s + 25-s − 0.187·26-s + 4.06·27-s + 6.98·28-s + ⋯ |
| L(s) = 1 | − 0.408·2-s − 0.431·3-s − 0.833·4-s + 0.447·5-s + 0.176·6-s − 1.58·7-s + 0.748·8-s − 0.814·9-s − 0.182·10-s − 0.301·11-s + 0.359·12-s + 0.0899·13-s + 0.647·14-s − 0.192·15-s + 0.527·16-s − 1.21·17-s + 0.332·18-s − 0.229·19-s − 0.372·20-s + 0.683·21-s + 0.123·22-s + 1.69·23-s − 0.322·24-s + 0.200·25-s − 0.0367·26-s + 0.782·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5084405511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5084405511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.577T + 2T^{2} \) |
| 3 | \( 1 + 0.746T + 3T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 13 | \( 1 - 0.324T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 23 | \( 1 - 8.11T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 - 5.77T + 37T^{2} \) |
| 41 | \( 1 - 0.484T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 7.16T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 - 8.20T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 2.88T + 73T^{2} \) |
| 79 | \( 1 - 3.95T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.705288066699884810052073533056, −9.161852408757785521096469987063, −8.626944333041461485565759572176, −7.35844696984305621599424300303, −6.46148408184673605233423716421, −5.71166660061411431711770384942, −4.81128232497304778083778848641, −3.61856112149733872448809411087, −2.54867269444846604948318581572, −0.57428425099172084936328646666,
0.57428425099172084936328646666, 2.54867269444846604948318581572, 3.61856112149733872448809411087, 4.81128232497304778083778848641, 5.71166660061411431711770384942, 6.46148408184673605233423716421, 7.35844696984305621599424300303, 8.626944333041461485565759572176, 9.161852408757785521096469987063, 9.705288066699884810052073533056
