L(s) = 1 | + 2.66·2-s − 3-s + 5.10·4-s + (1.18 + 1.89i)5-s − 2.66·6-s + (−2.09 + 1.61i)7-s + 8.26·8-s + 9-s + (3.15 + 5.05i)10-s + (1.90 − 2.71i)11-s − 5.10·12-s − 0.864i·13-s + (−5.59 + 4.29i)14-s + (−1.18 − 1.89i)15-s + 11.8·16-s − 1.53i·17-s + ⋯ |
L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.55·4-s + (0.530 + 0.847i)5-s − 1.08·6-s + (−0.793 + 0.608i)7-s + 2.92·8-s + 0.333·9-s + (0.998 + 1.59i)10-s + (0.573 − 0.819i)11-s − 1.47·12-s − 0.239i·13-s + (−1.49 + 1.14i)14-s + (−0.306 − 0.489i)15-s + 2.95·16-s − 0.372i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.627997614\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.627997614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (-1.18 - 1.89i)T \) |
| 7 | \( 1 + (2.09 - 1.61i)T \) |
| 11 | \( 1 + (-1.90 + 2.71i)T \) |
good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 13 | \( 1 + 0.864iT - 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 - 7.32iT - 23T^{2} \) |
| 29 | \( 1 + 1.42iT - 29T^{2} \) |
| 31 | \( 1 - 0.551iT - 31T^{2} \) |
| 37 | \( 1 - 5.97iT - 37T^{2} \) |
| 41 | \( 1 + 0.607T + 41T^{2} \) |
| 43 | \( 1 - 0.823T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 + 8.21iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 6.95T + 61T^{2} \) |
| 67 | \( 1 + 7.78iT - 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 3.89iT - 73T^{2} \) |
| 79 | \( 1 + 8.94iT - 79T^{2} \) |
| 83 | \( 1 + 13.1iT - 83T^{2} \) |
| 89 | \( 1 - 7.51iT - 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−10.13034923787874906386282033914, −9.370643578742214497417790672320, −7.74264808106539888677842312050, −6.77552095564124192971738511855, −6.26343297191658783203809129351, −5.65852132191192773805746259602, −4.93051575683780321576142005587, −3.40574829397977887947436075468, −3.21421486760847085076332318348, −1.83212157458443855927858798318,
1.36093657650657322727647258888, 2.62709507507837233474364319161, 3.99332051920652785762054440260, 4.45030437259874657582132497806, 5.35339617981946979428517846229, 6.18173626654563571179947681144, 6.73619933684709711618019323817, 7.56210429189727325450272157925, 9.047734317927968619327934569726, 10.05626460109101746884372987156