| L(s) = 1 | + i·5-s − 1.41i·7-s + 3.84·11-s + 0.406·13-s − 4.43i·17-s + 5.26i·19-s − 0.992·23-s − 25-s + 8.26i·29-s + 5.00i·31-s + 1.41·35-s − 1.01·37-s + 7.68i·41-s − 1.42i·43-s − 6.66·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.534i·7-s + 1.15·11-s + 0.112·13-s − 1.07i·17-s + 1.20i·19-s − 0.206·23-s − 0.200·25-s + 1.53i·29-s + 0.899i·31-s + 0.239·35-s − 0.167·37-s + 1.19i·41-s − 0.217i·43-s − 0.972·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.935262429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935262429\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 - 0.406T + 13T^{2} \) |
| 17 | \( 1 + 4.43iT - 17T^{2} \) |
| 19 | \( 1 - 5.26iT - 19T^{2} \) |
| 23 | \( 1 + 0.992T + 23T^{2} \) |
| 29 | \( 1 - 8.26iT - 29T^{2} \) |
| 31 | \( 1 - 5.00iT - 31T^{2} \) |
| 37 | \( 1 + 1.01T + 37T^{2} \) |
| 41 | \( 1 - 7.68iT - 41T^{2} \) |
| 43 | \( 1 + 1.42iT - 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 1.78iT - 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 8.61iT - 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 - 8.64iT - 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 3.19iT - 89T^{2} \) |
| 97 | \( 1 - 0.575T + 97T^{2} \) |
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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
−8.187132085102598372539747305338, −7.40480388130110402717712203958, −6.79535746465677256508353651576, −6.28783637472800079948092977035, −5.31760840897377341904136826785, −4.56008955312286667902301285958, −3.62990696459947981324741045438, −3.18107604795620104544449865153, −1.89198930328465645611993647098, −1.01889079392255972421650571655,
0.57793009697547327034233956706, 1.73908483485959184309806294214, 2.53506553028794068768609894705, 3.73856904182080899312895603070, 4.24138894616464357803984137741, 5.13135057353188438613427358824, 5.99399469422998885256287025603, 6.41633117711018198410003428534, 7.31762532780615651896157468444, 8.131777506029346176066378683620
