L(s) = 1 | + 1.54i·2-s − i·3-s − 0.393·4-s − i·5-s + 1.54·6-s + (2.51 + 0.814i)7-s + 2.48i·8-s − 9-s + 1.54·10-s + (1.71 − 2.83i)11-s + 0.393i·12-s + 2.83·13-s + (−1.26 + 3.89i)14-s − 15-s − 4.63·16-s + 1.11·17-s + ⋯ |
L(s) = 1 | + 1.09i·2-s − 0.577i·3-s − 0.196·4-s − 0.447i·5-s + 0.631·6-s + (0.951 + 0.308i)7-s + 0.878i·8-s − 0.333·9-s + 0.489·10-s + (0.518 − 0.855i)11-s + 0.113i·12-s + 0.785·13-s + (−0.336 + 1.04i)14-s − 0.258·15-s − 1.15·16-s + 0.269·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.111164699\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.111164699\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.51 - 0.814i)T \) |
| 11 | \( 1 + (-1.71 + 2.83i)T \) |
good | 2 | \( 1 - 1.54iT - 2T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 + 2.50iT - 29T^{2} \) |
| 31 | \( 1 + 4.63iT - 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 3.19iT - 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 - 4.55iT - 59T^{2} \) |
| 61 | \( 1 - 2.53T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 4.77iT - 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 0.588iT - 89T^{2} \) |
| 97 | \( 1 + 7.73iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.536688448435642775893316664282, −8.645440266874491958566299750788, −8.074564230537139479532109340806, −7.59672822986723664808740288840, −6.38934679435030758389849158518, −5.89614334985202569774908229930, −5.12690479524490834533124144581, −3.93521800142714244141348470775, −2.40302380448107986376230500420, −1.17486902581707544379442531831,
1.26615352488553429889154016573, 2.28554483733400494354672459884, 3.52206357655019473920296189905, 4.12744352419009109408379647027, 5.15544757338356881057978646966, 6.37289415304614849859480762505, 7.25656332046411284781884267113, 8.171477762284410864040558296822, 9.257134073497488367945304081628, 9.903686776629468658798908594727