| L(s) = 1 | + 1.77i·5-s + (2.39 − 1.12i)7-s + 3.85·11-s − 6.59·13-s − 4.56·17-s + 1.05·19-s + 0.0946i·23-s + 1.83·25-s + 4.31·29-s − 4.07i·31-s + (2.00 + 4.25i)35-s + 4.65i·37-s + 11.5·41-s + 6.28i·43-s + 6.12·47-s + ⋯ |
| L(s) = 1 | + 0.794i·5-s + (0.904 − 0.426i)7-s + 1.16·11-s − 1.82·13-s − 1.10·17-s + 0.241·19-s + 0.0197i·23-s + 0.367·25-s + 0.801·29-s − 0.732i·31-s + (0.339 + 0.718i)35-s + 0.764i·37-s + 1.80·41-s + 0.958i·43-s + 0.894·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.020065710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020065710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.39 + 1.12i)T \) |
| good | 5 | \( 1 - 1.77iT - 5T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.05T + 19T^{2} \) |
| 23 | \( 1 - 0.0946iT - 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 + 4.07iT - 31T^{2} \) |
| 37 | \( 1 - 4.65iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 6.28iT - 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 - 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 4.83iT - 67T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 2.25iT - 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 10.6iT - 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−8.552458769504380721953818537620, −7.56517995527232923261394122016, −7.15543304860949117094527596866, −6.51964104855030502029133110131, −5.57789260156332909018517785301, −4.46504300364415363783203711773, −4.28036506673304919909347557843, −2.87649135147550564199856249025, −2.24419767626100396071590343274, −0.992904994474181253753391315865,
0.69824798228243522249757482403, 1.86764832877942099393511242707, 2.60533733366390306439978526481, 3.97677365482960367273028266811, 4.74252665832051359668882511978, 5.09172954752869486299924218469, 6.10517789303297940158030796338, 7.00418491175071750898940543540, 7.59050562482640268902610816374, 8.522199891703776736696772771727
