L(s) = 1 | + (1.89 − 1.18i)5-s − 0.992i·7-s − 1.83·11-s − 3.28i·13-s + 6.63i·17-s + 5.64·19-s − i·23-s + (2.20 − 4.48i)25-s + 2.01·29-s − 0.315·31-s + (−1.17 − 1.88i)35-s − 3.07i·37-s + 1.34·41-s + 5.97i·43-s − 0.306i·47-s + ⋯ |
L(s) = 1 | + (0.848 − 0.528i)5-s − 0.375i·7-s − 0.552·11-s − 0.911i·13-s + 1.60i·17-s + 1.29·19-s − 0.208i·23-s + (0.441 − 0.897i)25-s + 0.374·29-s − 0.0565·31-s + (−0.198 − 0.318i)35-s − 0.505i·37-s + 0.210·41-s + 0.911i·43-s − 0.0446i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.225823361\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.225823361\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.89 + 1.18i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + 0.992iT - 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 + 3.28iT - 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 + 0.315T + 31T^{2} \) |
| 37 | \( 1 + 3.07iT - 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 - 5.97iT - 43T^{2} \) |
| 47 | \( 1 + 0.306iT - 47T^{2} \) |
| 53 | \( 1 + 6.98iT - 53T^{2} \) |
| 59 | \( 1 - 9.49T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 0.853iT - 67T^{2} \) |
| 71 | \( 1 + 0.797T + 71T^{2} \) |
| 73 | \( 1 + 7.67iT - 73T^{2} \) |
| 79 | \( 1 - 3.62T + 79T^{2} \) |
| 83 | \( 1 + 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 18.5iT - 97T^{2} \) |
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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
−8.269270215555299510096006497703, −7.71780636998516491695102552524, −6.79614609452166813282137725691, −5.89020885191535910840954124954, −5.46912457265017106614451880134, −4.63560423922238075221676416707, −3.67321544230038194604132478376, −2.75012588504296961430116435233, −1.71806150941575230178112999895, −0.71839001199297752460081920170,
1.09143860345320873822987730507, 2.35282851102958357186560571714, 2.81776971662551377573540692884, 3.89704619389198376227883564272, 5.15942978474675230244724478772, 5.36019895264590474985489691196, 6.41117708775074733044182230004, 7.07126536324437356305845100872, 7.61093270428895817097063893470, 8.705521573236313620265916563300