L(s) = 1 | − 2-s + (−0.707 + 1.58i)3-s + 4-s + (−1.41 + 1.73i)5-s + (0.707 − 1.58i)6-s − 8-s + (−2.00 − 2.23i)9-s + (1.41 − 1.73i)10-s + 0.213i·11-s + (−0.707 + 1.58i)12-s − 6.70·13-s + (−1.73 − 3.46i)15-s + 16-s + 3.16i·17-s + (2.00 + 2.23i)18-s − 4.89i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 + 0.912i)3-s + 0.5·4-s + (−0.632 + 0.774i)5-s + (0.288 − 0.645i)6-s − 0.353·8-s + (−0.666 − 0.745i)9-s + (0.447 − 0.547i)10-s + 0.0643i·11-s + (−0.204 + 0.456i)12-s − 1.85·13-s + (−0.448 − 0.893i)15-s + 0.250·16-s + 0.766i·17-s + (0.471 + 0.527i)18-s − 1.12i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4304927780\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4304927780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 - 1.58i)T \) |
| 5 | \( 1 + (1.41 - 1.73i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.213iT - 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 - 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 2.02iT - 29T^{2} \) |
| 31 | \( 1 + 0.301iT - 31T^{2} \) |
| 37 | \( 1 + 7.13iT - 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 - 7.75iT - 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 5.32iT - 67T^{2} \) |
| 71 | \( 1 - 2.02iT - 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.522T + 79T^{2} \) |
| 83 | \( 1 + 16.7iT - 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−9.541408951443211354773153675379, −8.864293295860972096578557042607, −7.81976694556416250599512505482, −7.09660558275027293934765311267, −6.39724210551453337274760653137, −5.19792780618778303709869400632, −4.43174135214387590500695416833, −3.27498190237387852867636385627, −2.44808239531683371584985846032, −0.30162534940911610021189028545,
0.880230480803108207190586077926, 2.08552017472706229988351810647, 3.25251039690017186892125516797, 4.88558995478561404971931586549, 5.30420158131283596306915736117, 6.65878444400640278716505812493, 7.26454804207265320604478857095, 7.903933321329991070834504005507, 8.600589615001013067012675138748, 9.473171598108962458228018595197