L(s) = 1 | + 2.41i·2-s − 3.82·4-s − 0.414i·7-s − 4.41i·8-s + 11-s − 2.82i·13-s + 0.999·14-s + 2.99·16-s + 2.41i·17-s − 6.41·19-s + 2.41i·22-s − i·23-s + 6.82·26-s + 1.58i·28-s + 1.17·29-s + ⋯ |
L(s) = 1 | + 1.70i·2-s − 1.91·4-s − 0.156i·7-s − 1.56i·8-s + 0.301·11-s − 0.784i·13-s + 0.267·14-s + 0.749·16-s + 0.585i·17-s − 1.47·19-s + 0.514i·22-s − 0.208i·23-s + 1.33·26-s + 0.299i·28-s + 0.217·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9732440559\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9732440559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 7 | \( 1 + 0.414iT - 7T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 2.41iT - 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 0.171iT - 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 11.6iT - 43T^{2} \) |
| 47 | \( 1 + 7.48iT - 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 0.343iT - 67T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 + 8.82iT - 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.48iT - 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 5.82iT - 97T^{2} \) |
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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
−8.569088326392266664379366328814, −8.273300086561140657440185602871, −7.22541549071877945912088325647, −6.85064998890396923875788243425, −5.83864675941068292794131522268, −5.47793647540764574041079792548, −4.34850907095944858481120054710, −3.74418375022745920690178854001, −2.17016719000257800325306429748, −0.36380217726064410013608527407,
1.12078926378051411794122827619, 2.15509539755876009654417942676, 2.85454320349412162939625092790, 4.05294231511408836867187843073, 4.36931673810985862742300796462, 5.52153634804185277992862589116, 6.51015314469844169360299720441, 7.45586358620511501456185291098, 8.551137181817035663919411156441, 9.164252935011949179778831236172