| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2.82i·7-s + i·8-s − 9-s + 5.65·11-s − i·12-s − i·13-s + 2.82·14-s + 16-s − 0.828i·17-s + i·18-s − 2.82·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.06i·7-s + 0.353i·8-s − 0.333·9-s + 1.70·11-s − 0.288i·12-s − 0.277i·13-s + 0.755·14-s + 0.250·16-s − 0.200i·17-s + 0.235i·18-s − 0.648·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730580954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730580954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 6.34iT - 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 3.17iT - 97T^{2} \) |
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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
−9.214117514417822632078334322867, −8.709966062975001496905949412974, −8.121942755762354301325241373313, −6.43640408334463485610490584492, −6.25937630540701701760101101182, −4.78491858075126431032517320221, −4.45144669092890016851131729770, −3.21557648644940086338065103100, −2.50691721364991197265604180035, −1.15105318031129360090332923900,
0.76825043146182310805696085172, 1.85060082777297354029388737562, 3.62452457784166558281009349183, 4.07330498168643432907503184240, 5.23232672530657350033399889749, 6.21796282018900067536929243605, 6.93904326375514261259502982522, 7.22384939933588889454988843750, 8.356438626793956529147882326267, 8.862412976972697615857456152954
