L(s) = 1 | + (−1 − i)2-s + (−0.5 + 0.133i)3-s + 2i·4-s + (−0.133 − 2.23i)5-s + (0.633 + 0.366i)6-s + (−2.5 + 0.866i)7-s + (2 − 2i)8-s + (−2.36 + 1.36i)9-s + (−2.09 + 2.36i)10-s + (2.36 + 1.36i)11-s + (−0.267 − i)12-s + (−3 − 3i)13-s + (3.36 + 1.63i)14-s + (0.366 + 1.09i)15-s − 4·16-s + (−4.09 + 1.09i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.288 + 0.0773i)3-s + i·4-s + (−0.0599 − 0.998i)5-s + (0.258 + 0.149i)6-s + (−0.944 + 0.327i)7-s + (0.707 − 0.707i)8-s + (−0.788 + 0.455i)9-s + (−0.663 + 0.748i)10-s + (0.713 + 0.411i)11-s + (−0.0773 − 0.288i)12-s + (−0.832 − 0.832i)13-s + (0.899 + 0.436i)14-s + (0.0945 + 0.283i)15-s − 16-s + (−0.993 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (0.133 + 2.23i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 - 0.133i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 1.36i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.09 - 1.09i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 5.59i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + (0.169 + 0.0980i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.464 + 1.73i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 6.46iT - 41T^{2} \) |
| 43 | \( 1 + (0.633 - 0.633i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.36 + 8.83i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3 + 11.1i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (4.26 + 2.46i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.96 + 3.40i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.96 - 2.13i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 8.53T + 71T^{2} \) |
| 73 | \( 1 + (2.90 + 10.8i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.803 - 0.464i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.56 - 7.56i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.40 + 5.89i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.73 - 9.73i)T + 97iT^{2} \) |
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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
−11.35047523241033052477368768580, −10.18581632910974073149982339875, −9.411702831550052466275725029017, −8.638380396110058048965093890751, −7.67743553580030210321353504033, −6.27963831145766162924715230676, −4.97854142869800318530993248435, −3.64600100804541746265500342734, −2.11489822527511908787053687211, 0,
2.59475920710829249919940461625, 4.29332914125950315377178814389, 6.08168498421144830718976732131, 6.55945260221215045398144786333, 7.31159751016880320366546950420, 8.809742612845560140779735292210, 9.427123086148708300677871491120, 10.55522934305167770215341721720, 11.21490028457205264367185592494