| L(s) = 1 | − 2.47·3-s + 2.59·5-s + 7-s + 3.12·9-s − 4.33·11-s − 4.86·13-s − 6.42·15-s − 5.17·17-s − 7.11·19-s − 2.47·21-s + 2.29·23-s + 1.74·25-s − 0.314·27-s + 8.20·29-s − 1.04·31-s + 10.7·33-s + 2.59·35-s + 4.77·37-s + 12.0·39-s + 7.95·41-s − 8.63·43-s + 8.12·45-s + 6.55·47-s + 49-s + 12.8·51-s + 10.3·53-s − 11.2·55-s + ⋯ |
| L(s) = 1 | − 1.42·3-s + 1.16·5-s + 0.377·7-s + 1.04·9-s − 1.30·11-s − 1.34·13-s − 1.65·15-s − 1.25·17-s − 1.63·19-s − 0.540·21-s + 0.479·23-s + 0.348·25-s − 0.0606·27-s + 1.52·29-s − 0.187·31-s + 1.86·33-s + 0.438·35-s + 0.784·37-s + 1.92·39-s + 1.24·41-s − 1.31·43-s + 1.21·45-s + 0.956·47-s + 0.142·49-s + 1.79·51-s + 1.41·53-s − 1.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8709488526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8709488526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 + 7.11T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 4.77T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 + 8.63T + 43T^{2} \) |
| 47 | \( 1 - 6.55T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 3.18T + 83T^{2} \) |
| 89 | \( 1 - 8.12T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
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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
−8.589767265608973886836510080663, −7.70194914055468024146966434691, −6.75118984420100058964420363060, −6.29443509479835077747835856544, −5.51207691948698389869704816619, −4.88629064777092129358475513108, −4.44778501292860031509534221394, −2.56306931306227628066157679779, −2.12327038888976119076027789085, −0.55927215732187318767040243737,
0.55927215732187318767040243737, 2.12327038888976119076027789085, 2.56306931306227628066157679779, 4.44778501292860031509534221394, 4.88629064777092129358475513108, 5.51207691948698389869704816619, 6.29443509479835077747835856544, 6.75118984420100058964420363060, 7.70194914055468024146966434691, 8.589767265608973886836510080663
