L(s) = 1 | − i·2-s − 3-s − 4-s + i·5-s + i·6-s + 2i·7-s + i·8-s + 9-s + 10-s + i·11-s + 12-s + 2·14-s − i·15-s + 16-s − 2·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 0.755i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.301i·11-s + 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s − 0.485·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.254697384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.254697384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 3iT - 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + 5iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.349473859479836628470175526276, −7.23737362341621947535403576022, −6.82400945157533984914835099993, −5.84042294540005821525179389638, −5.19341971990176875015362820515, −4.49918098312828972745521688840, −3.56059420124176434576397231008, −2.61956663383141666578452604900, −1.97122151110831402224764351116, −0.60179568925753632161282610233,
0.67003337186455975986198714174, 1.65360362690956341577402281819, 3.18601641569414944243713808471, 4.12672951268735207937340952666, 4.64005238413911631868106589778, 5.54895163414580302947710408022, 6.07817884334983330702626992726, 6.87456564460556029096669783655, 7.47771080534668747785694431991, 8.233431023005008434394608399032