L(s) = 1 | + (1.66 − 1.66i)3-s + (2.27 − 1.34i)7-s − 2.56i·9-s + 1.56·11-s + (2.60 − 2.60i)13-s + (2.60 + 2.60i)17-s − 2.64·19-s + (1.56 − 6.04i)21-s + (−0.794 − 0.794i)23-s + (0.731 + 0.731i)27-s − 6.68i·29-s + 9.43i·31-s + (2.60 − 2.60i)33-s + (−2.82 + 2.82i)37-s − 8.68i·39-s + ⋯ |
L(s) = 1 | + (0.962 − 0.962i)3-s + (0.861 − 0.507i)7-s − 0.853i·9-s + 0.470·11-s + (0.722 − 0.722i)13-s + (0.631 + 0.631i)17-s − 0.607·19-s + (0.340 − 1.31i)21-s + (−0.165 − 0.165i)23-s + (0.140 + 0.140i)27-s − 1.24i·29-s + 1.69i·31-s + (0.453 − 0.453i)33-s + (−0.464 + 0.464i)37-s − 1.39i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.715162825\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.715162825\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.27 + 1.34i)T \) |
good | 3 | \( 1 + (-1.66 + 1.66i)T - 3iT^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + (-2.60 + 2.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.60 - 2.60i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + (0.794 + 0.794i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.68iT - 29T^{2} \) |
| 31 | \( 1 - 9.43iT - 31T^{2} \) |
| 37 | \( 1 + (2.82 - 2.82i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.64iT - 41T^{2} \) |
| 43 | \( 1 + (6.45 + 6.45i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.66 - 1.66i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.24 - 7.24i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 9.43iT - 61T^{2} \) |
| 67 | \( 1 + (-3.62 + 3.62i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + (6.67 - 6.67i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (9.47 - 9.47i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 + (5.52 + 5.52i)T + 97iT^{2} \) |
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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
−9.083499350847383561467002658323, −8.305036299934197856785063366379, −8.020172123728555761574065875363, −7.11350461177578592714652341705, −6.33935541742697661788749717993, −5.24751090484326628025756261650, −4.06065836267045726538688970266, −3.19425644083610355949242626304, −1.95806496177385972264585853452, −1.13910070527970706812104428059,
1.60142988333774182713016272575, 2.74307097029771565760046149994, 3.77725205314627194403071516401, 4.44748557832365193096283561931, 5.38667999701862761714694481578, 6.41961675211597587866568203904, 7.53986605987736425295188604915, 8.413514683412712572941438486978, 8.913832909539760183284091876926, 9.543933443981995960140458355785