| L(s) = 1 | + 1.30·2-s + 3-s − 0.302·4-s + 1.30·6-s − 3·8-s + 9-s − 3·11-s − 0.302·12-s + 4.60·13-s − 3.30·16-s − 2.60·17-s + 1.30·18-s + 0.605·19-s − 3.90·22-s − 8.21·23-s − 3·24-s + 6·26-s + 27-s − 0.394·29-s − 7.21·31-s + 1.69·32-s − 3·33-s − 3.39·34-s − 0.302·36-s − 10.2·37-s + 0.788·38-s + 4.60·39-s + ⋯ |
| L(s) = 1 | + 0.921·2-s + 0.577·3-s − 0.151·4-s + 0.531·6-s − 1.06·8-s + 0.333·9-s − 0.904·11-s − 0.0874·12-s + 1.27·13-s − 0.825·16-s − 0.631·17-s + 0.307·18-s + 0.138·19-s − 0.833·22-s − 1.71·23-s − 0.612·24-s + 1.17·26-s + 0.192·27-s − 0.0732·29-s − 1.29·31-s + 0.300·32-s − 0.522·33-s − 0.582·34-s − 0.0504·36-s − 1.67·37-s + 0.127·38-s + 0.737·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 + 8.21T + 23T^{2} \) |
| 29 | \( 1 + 0.394T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 6.60T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.275578923869579581678649967951, −7.44613662530466639315094134783, −6.48925868697257161266471191489, −5.74995990895694318809327625707, −5.13421765817438793379118565886, −4.08804768540649445920529278707, −3.68896079250887193015175359670, −2.75450300721652680578859920863, −1.76274702040264180908585726671, 0,
1.76274702040264180908585726671, 2.75450300721652680578859920863, 3.68896079250887193015175359670, 4.08804768540649445920529278707, 5.13421765817438793379118565886, 5.74995990895694318809327625707, 6.48925868697257161266471191489, 7.44613662530466639315094134783, 8.275578923869579581678649967951
