| L(s) = 1 | − 3-s − 0.765i·5-s + 9-s + 2.16i·11-s − 0.317i·13-s + 0.765i·15-s + 3.37i·17-s − 5.65·19-s − 5.22i·23-s + 4.41·25-s − 27-s − 2.58·29-s − 1.65·31-s − 2.16i·33-s − 1.41·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.342i·5-s + 0.333·9-s + 0.652i·11-s − 0.0879i·13-s + 0.197i·15-s + 0.819i·17-s − 1.29·19-s − 1.08i·23-s + 0.882·25-s − 0.192·27-s − 0.480·29-s − 0.297·31-s − 0.376i·33-s − 0.232·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2373732264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2373732264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.765iT - 5T^{2} \) |
| 11 | \( 1 - 2.16iT - 11T^{2} \) |
| 13 | \( 1 + 0.317iT - 13T^{2} \) |
| 17 | \( 1 - 3.37iT - 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.22iT - 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 5.99iT - 41T^{2} \) |
| 43 | \( 1 - 7.39iT - 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 7.07iT - 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 6.62iT - 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 3.82iT - 97T^{2} \) |
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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
−9.264597675562794197444250606619, −8.554860304282381799423874785429, −7.83411836448650630283302380449, −6.79227115127176918896744241328, −6.32647727843282047794853010092, −5.32343922505680147297635481476, −4.57977351298623338823676199041, −3.86233774080046789716250612311, −2.48449953552213712499924520481, −1.42291480824776091817026402485,
0.087464364595941041859177282238, 1.56566591874379821537916830576, 2.82379802445324777454984750692, 3.74671946656088102014515150209, 4.76040216370376423020191417908, 5.52463584195135138291247305371, 6.35828294020820344739970019891, 6.98139376158271156703835583670, 7.81080316838531594192051488428, 8.698958667879034007078690308493
