| L(s) = 1 | − 1.66i·2-s − 1.15i·3-s − 0.769·4-s + (1.63 + 1.52i)5-s − 1.92·6-s + 3.92i·7-s − 2.04i·8-s + 1.66·9-s + (2.53 − 2.72i)10-s − 11-s + 0.889i·12-s + 3.15i·13-s + 6.53·14-s + (1.76 − 1.89i)15-s − 4.94·16-s + 7.92i·17-s + ⋯ |
| L(s) = 1 | − 1.17i·2-s − 0.667i·3-s − 0.384·4-s + (0.731 + 0.682i)5-s − 0.785·6-s + 1.48i·7-s − 0.723i·8-s + 0.554·9-s + (0.802 − 0.860i)10-s − 0.301·11-s + 0.256i·12-s + 0.874i·13-s + 1.74·14-s + (0.455 − 0.488i)15-s − 1.23·16-s + 1.92i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.997017910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.997017910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.63 - 1.52i)T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.66iT - 2T^{2} \) |
| 3 | \( 1 + 1.15iT - 3T^{2} \) |
| 7 | \( 1 - 3.92iT - 7T^{2} \) |
| 13 | \( 1 - 3.15iT - 13T^{2} \) |
| 17 | \( 1 - 7.92iT - 17T^{2} \) |
| 23 | \( 1 - 0.0689iT - 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 - 3.48iT - 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 + 6.72iT - 43T^{2} \) |
| 47 | \( 1 + 8.28iT - 47T^{2} \) |
| 53 | \( 1 - 4.07iT - 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 + 3.16iT - 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 + 9.78iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 5.79iT - 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−10.02573995735241453771603327729, −9.226386483426556597263007840174, −8.380585478829485032973635844862, −7.15076917309137468374301694369, −6.38568587075879049178519082542, −5.73776349993615370286159545769, −4.26927773866159576849358551703, −3.06855891144013441961793824030, −2.10399493430740839170494420070, −1.65767424419740043830627963265,
0.954625702943947213392610467028, 2.78380646858355416236743325967, 4.30222193743264081017441843376, 4.91703809981470020371207025038, 5.66342248137842233562276534132, 6.79325615515135022143925031482, 7.42902812139936556213444630735, 8.114807798348265820301729258433, 9.288444199545974435746624583536, 9.823071063660197422122149971846
