L(s) = 1 | − 2.33i·2-s − 1.97i·3-s − 3.44·4-s + (−1.75 − 1.39i)5-s − 4.60·6-s + 4.79i·7-s + 3.37i·8-s − 0.889·9-s + (−3.24 + 4.08i)10-s + 11-s + 6.79i·12-s + 6.97i·13-s + 11.1·14-s + (−2.74 + 3.45i)15-s + 0.985·16-s − 1.90i·17-s + ⋯ |
L(s) = 1 | − 1.65i·2-s − 1.13i·3-s − 1.72·4-s + (−0.783 − 0.621i)5-s − 1.87·6-s + 1.81i·7-s + 1.19i·8-s − 0.296·9-s + (−1.02 + 1.29i)10-s + 0.301·11-s + 1.96i·12-s + 1.93i·13-s + 2.98·14-s + (−0.708 + 0.891i)15-s + 0.246·16-s − 0.461i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6771173295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6771173295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.75 + 1.39i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 2.33iT - 2T^{2} \) |
| 3 | \( 1 + 1.97iT - 3T^{2} \) |
| 7 | \( 1 - 4.79iT - 7T^{2} \) |
| 13 | \( 1 - 6.97iT - 13T^{2} \) |
| 17 | \( 1 + 1.90iT - 17T^{2} \) |
| 23 | \( 1 - 4.07iT - 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 5.25iT - 37T^{2} \) |
| 41 | \( 1 + 4.11T + 41T^{2} \) |
| 43 | \( 1 - 2.83iT - 43T^{2} \) |
| 47 | \( 1 - 9.58iT - 47T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 - 3.08T + 59T^{2} \) |
| 61 | \( 1 + 5.73T + 61T^{2} \) |
| 67 | \( 1 + 5.17iT - 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 - 2.32iT - 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 2.49iT - 83T^{2} \) |
| 89 | \( 1 + 8.26T + 89T^{2} \) |
| 97 | \( 1 + 0.339iT - 97T^{2} \) |
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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
−9.526009767711792488729865336403, −9.228407756466503695506422148728, −8.533787542139354486127676354079, −7.46522176243418205866393470958, −6.48811730477252374785820741405, −5.26752841774101690697257273305, −4.31323254227256208878583317765, −3.21698082401179116172098218400, −2.01189964434016569447052999423, −1.50555830173936986421068117664,
0.31753370506964171560761569750, 3.49991200246673469304528492769, 3.93686050317386396778982415154, 4.84191721527761855220360573541, 5.74210120406686168494619016204, 6.91772457538705970328618183570, 7.40921859030110940284593777861, 8.031352345949509581918805470771, 8.965810813538767096120874703525, 10.12250547732720681516292103220