L(s) = 1 | − 2-s + 4-s + (2.23 + 1.41i)7-s − 8-s + 1.41i·11-s + 5.39·13-s + (−2.23 − 1.41i)14-s + 16-s + 2.23i·17-s − 1.30i·19-s − 1.41i·22-s − 23-s − 5.39·26-s + (2.23 + 1.41i)28-s + 9.24i·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.845 + 0.534i)7-s − 0.353·8-s + 0.426i·11-s + 1.49·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s + 0.542i·17-s − 0.300i·19-s − 0.301i·22-s − 0.208·23-s − 1.05·26-s + (0.422 + 0.267i)28-s + 1.71i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.506887979\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.506887979\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.23 - 1.41i)T \) |
good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 + 1.30iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 9.24iT - 29T^{2} \) |
| 31 | \( 1 - 8.56iT - 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 + 6.41iT - 43T^{2} \) |
| 47 | \( 1 + 7.63iT - 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.39iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4.34iT - 71T^{2} \) |
| 73 | \( 1 - 5.01T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 8.01iT - 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.644115147538370454275037459634, −8.408905125988366412537697382565, −7.37661171790404812779295917248, −6.73464609885703768456072381838, −5.80876814000058531693700558426, −5.14954079692918725464441484334, −4.06907688792391475370019832497, −3.12440634316891798124181334725, −1.93633265305773165420542068486, −1.21819689321840552581469706288,
0.66876321119493066531613599678, 1.59567589643450197558645628472, 2.72796970284675827340773650657, 3.85336897267693793237787857978, 4.53058625399700144637155248258, 5.82005755491265537459993612372, 6.20168892471255021340612162142, 7.28279984870819510406459061734, 8.037170477523292035786303089548, 8.305204230031972119231515587163