| L(s) = 1 | − i·5-s − 2.44i·7-s + 5.91·11-s − 6.24·13-s − 4.89i·19-s − 1.43·23-s − 25-s − 6i·29-s − 1.43i·31-s − 2.44·35-s − 2.24·37-s − 4.24i·41-s − 11.8i·43-s + 11.8·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.925i·7-s + 1.78·11-s − 1.73·13-s − 1.12i·19-s − 0.299·23-s − 0.200·25-s − 1.11i·29-s − 0.257i·31-s − 0.414·35-s − 0.368·37-s − 0.662i·41-s − 1.80i·43-s + 1.72·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01575 - 0.926461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01575 - 0.926461i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 1.43iT - 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 11.8iT - 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.48iT - 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 8.36iT - 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 7.75iT - 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−10.06040228154270468305063686353, −9.398886462450061172305109046518, −8.641133829690049550796154558230, −7.31126283088763246798646711314, −6.98702615302102325612927607826, −5.70251006352394927166266985504, −4.51050464307576539546563790678, −3.91982076917214640012925954788, −2.30078138460316455810046653162, −0.73467324122291202174693202490,
1.74256130188814205139777465577, 2.95793744048484430582201834480, 4.12382324741991683050595921203, 5.26753568097859381559154418844, 6.26402495890623251663973043192, 7.02454336326630158777400442915, 8.017390885042379715151985423581, 9.094240310766804606828842833890, 9.618865016098485030757221078776, 10.53192016702995829092468266489
