| L(s) = 1 | − 1.41i·5-s + 4i·7-s + 5.65·11-s + 4·13-s − 4.24i·17-s − 5.65·23-s + 2.99·25-s + 1.41i·29-s + 4i·31-s + 5.65·35-s − 6·37-s + 9.89i·41-s − 8i·43-s − 5.65·47-s − 9·49-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s + 1.51i·7-s + 1.70·11-s + 1.10·13-s − 1.02i·17-s − 1.17·23-s + 0.599·25-s + 0.262i·29-s + 0.718i·31-s + 0.956·35-s − 0.986·37-s + 1.54i·41-s − 1.21i·43-s − 0.825·47-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39311 + 0.118643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39311 + 0.118643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−11.98500951160907510464449286769, −11.16645002588220080274442554202, −9.644126702829328364594173034632, −8.922146539341016817317765030684, −8.380563515705680089235661890331, −6.72782536425731826204413328996, −5.86877845674063734686819743619, −4.75008815234871692264760887553, −3.34668919802155883136794230939, −1.61484340692060727880892807346,
1.41187595928568032468506747292, 3.62885744796318735999549096368, 4.16004306840503940575054079219, 6.15045363800181020716343317172, 6.75756199129673234147427951280, 7.83883647137408232951340544555, 8.950571916396289238588122200600, 10.11769452306528091994413829570, 10.80713452116639014853300352022, 11.58952317778871482106020603340
