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
−11.21490028457205264367185592494, −10.55522934305167770215341721720, −9.427123086148708300677871491120, −8.809742612845560140779735292210, −7.31159751016880320366546950420, −6.55945260221215045398144786333, −6.08168498421144830718976732131, −4.29332914125950315377178814389, −2.59475920710829249919940461625, 0,
2.11489822527511908787053687211, 3.64600100804541746265500342734, 4.97854142869800318530993248435, 6.27963831145766162924715230676, 7.67743553580030210321353504033, 8.638380396110058048965093890751, 9.411702831550052466275725029017, 10.18581632910974073149982339875, 11.35047523241033052477368768580