L(s) = 1 | − 0.732i·2-s + 1.46·4-s − 4.73i·7-s − 2.53i·8-s + 5.73·11-s − 1.46i·13-s − 3.46·14-s + 1.07·16-s + 2.73i·17-s − 4.46·19-s − 4.19i·22-s − 3.46i·23-s − 1.07·26-s − 6.92i·28-s + 3.19·29-s + ⋯ |
L(s) = 1 | − 0.517i·2-s + 0.732·4-s − 1.78i·7-s − 0.896i·8-s + 1.72·11-s − 0.406i·13-s − 0.925·14-s + 0.267·16-s + 0.662i·17-s − 1.02·19-s − 0.894i·22-s − 0.722i·23-s − 0.210·26-s − 1.30i·28-s + 0.593·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.329221163\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.329221163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 0.732iT - 2T^{2} \) |
| 7 | \( 1 + 4.73iT - 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 2.73iT - 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 2.73iT - 37T^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 + 0.196iT - 43T^{2} \) |
| 47 | \( 1 - 8.73iT - 47T^{2} \) |
| 53 | \( 1 - 6.73iT - 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 - 7.66iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.19iT - 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + 9.66iT - 97T^{2} \) |
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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
−9.026744784726519087497594617186, −7.999796119894216333146818077731, −7.20905065089437835313981681639, −6.60413775222749163698229206804, −6.02265698693532791772044017157, −4.17678166205214135320890635040, −4.14369124628295645538307192909, −3.00273124164700206240772816749, −1.65930912530023178915196283739, −0.848246867776011159903908519155,
1.67071166874190612974472950840, 2.42905742205240507649360004244, 3.50497279141247677389639022210, 4.75152011131746763193572668082, 5.66119752295333949580766848680, 6.32550839166849040564627089699, 6.80399041775547669686780702238, 7.81845949848757794078266046776, 8.812086578354417372638567553362, 9.020616105888920011428479968997