| L(s) = 1 | + 1.90·3-s + 2.90·5-s − 7-s + 0.618·9-s + 2.79·11-s + 1.95·13-s + 5.52·15-s + 17-s − 4.42·19-s − 1.90·21-s + 1.72·23-s + 3.42·25-s − 4.53·27-s − 8.15·29-s + 9.64·31-s + 5.31·33-s − 2.90·35-s + 2.61·37-s + 3.71·39-s − 1.43·41-s + 8.99·43-s + 1.79·45-s + 2.65·47-s + 49-s + 1.90·51-s + 4.50·53-s + 8.10·55-s + ⋯ |
| L(s) = 1 | + 1.09·3-s + 1.29·5-s − 0.377·7-s + 0.206·9-s + 0.842·11-s + 0.541·13-s + 1.42·15-s + 0.242·17-s − 1.01·19-s − 0.415·21-s + 0.360·23-s + 0.684·25-s − 0.871·27-s − 1.51·29-s + 1.73·31-s + 0.925·33-s − 0.490·35-s + 0.429·37-s + 0.594·39-s − 0.224·41-s + 1.37·43-s + 0.267·45-s + 0.387·47-s + 0.142·49-s + 0.266·51-s + 0.618·53-s + 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.163583163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.163583163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 - 9.64T + 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 - 8.99T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 8.64T + 71T^{2} \) |
| 73 | \( 1 - 2.37T + 73T^{2} \) |
| 79 | \( 1 - 8.59T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.052339268933643340096031021568, −7.18617137714968631093070196980, −6.30742426914204626502146480120, −6.04408708699242834405831665720, −5.13195312169683749382178793611, −4.05129539922926690655285107763, −3.50815935918370658877403873936, −2.49840489940042788632842785416, −2.07153974051976643135584399764, −1.00029120283349766712854490917,
1.00029120283349766712854490917, 2.07153974051976643135584399764, 2.49840489940042788632842785416, 3.50815935918370658877403873936, 4.05129539922926690655285107763, 5.13195312169683749382178793611, 6.04408708699242834405831665720, 6.30742426914204626502146480120, 7.18617137714968631093070196980, 8.052339268933643340096031021568
