| L(s) = 1 | − 1.80·2-s + (−1.28 + 1.15i)3-s + 1.27·4-s + i·5-s + (2.32 − 2.09i)6-s − i·7-s + 1.31·8-s + (0.310 − 2.98i)9-s − 1.80i·10-s + (−1.44 + 2.98i)11-s + (−1.63 + 1.47i)12-s + 1.34i·13-s + 1.80i·14-s + (−1.15 − 1.28i)15-s − 4.92·16-s + 3.37·17-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + (−0.742 + 0.669i)3-s + 0.635·4-s + 0.447i·5-s + (0.949 − 0.856i)6-s − 0.377i·7-s + 0.466·8-s + (0.103 − 0.994i)9-s − 0.571i·10-s + (−0.434 + 0.900i)11-s + (−0.471 + 0.425i)12-s + 0.374i·13-s + 0.483i·14-s + (−0.299 − 0.332i)15-s − 1.23·16-s + 0.819·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2420250888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2420250888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 1.15i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (1.44 - 2.98i)T \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 13 | \( 1 - 1.34iT - 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 + 6.55iT - 19T^{2} \) |
| 23 | \( 1 - 0.712iT - 23T^{2} \) |
| 29 | \( 1 - 0.481T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 8.37iT - 43T^{2} \) |
| 47 | \( 1 - 8.28iT - 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 1.07iT - 59T^{2} \) |
| 61 | \( 1 + 9.12iT - 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 3.34iT - 71T^{2} \) |
| 73 | \( 1 + 8.92iT - 73T^{2} \) |
| 79 | \( 1 + 17.6iT - 79T^{2} \) |
| 83 | \( 1 - 0.931T + 83T^{2} \) |
| 89 | \( 1 - 5.94iT - 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−9.564394851501952659038836091751, −9.214123713414323876571251953264, −8.020624409943208577712082295610, −7.16309923554494988714642449466, −6.65435694555928988901786810332, −5.25877454560933769384955582917, −4.56085501643932530066424572904, −3.36802639060777760612226608781, −1.77738536396738607428681627525, −0.21957945232299169602892979278,
1.03970996463087667177919417298, 2.07255535067877010477533823709, 3.72193125603125680822902796074, 5.33904083730769029045619341239, 5.63781749385503789733633831244, 6.91172218829179830690186601725, 7.71836840524595888354918735883, 8.332237294376124739547914624051, 8.943399054536106521358642152643, 10.12340602762563989365129161555
