| L(s) = 1 | − 3.46·5-s − 1.41·7-s − 4.89·11-s + 1.41·13-s − 4.89·17-s + 6·19-s − 6.92·23-s + 6.99·25-s + 3.46·29-s + 1.41·31-s + 4.89·35-s − 9.89·37-s − 4.89·41-s + 6·43-s − 6.92·47-s − 5·49-s − 3.46·53-s + 16.9·55-s − 9.79·59-s − 7.07·61-s − 4.89·65-s − 8·67-s − 13.8·71-s − 12·73-s + 6.92·77-s − 15.5·79-s + 14.6·83-s + ⋯ |
| L(s) = 1 | − 1.54·5-s − 0.534·7-s − 1.47·11-s + 0.392·13-s − 1.18·17-s + 1.37·19-s − 1.44·23-s + 1.39·25-s + 0.643·29-s + 0.254·31-s + 0.828·35-s − 1.62·37-s − 0.765·41-s + 0.914·43-s − 1.01·47-s − 0.714·49-s − 0.475·53-s + 2.28·55-s − 1.27·59-s − 0.905·61-s − 0.607·65-s − 0.977·67-s − 1.64·71-s − 1.40·73-s + 0.789·77-s − 1.75·79-s + 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2442425263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2442425263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 9.79T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.63847141098529970365687474861, −7.28447967782485622908951424518, −6.41347043882812086622548276824, −5.67123945184384133351757331369, −4.74676703819523217533383360100, −4.30387988848262538151909509128, −3.24572694440190953289802574550, −3.01535763761885825616107218715, −1.71158503995434411321969205045, −0.22855338159461432523021291804,
0.22855338159461432523021291804, 1.71158503995434411321969205045, 3.01535763761885825616107218715, 3.24572694440190953289802574550, 4.30387988848262538151909509128, 4.74676703819523217533383360100, 5.67123945184384133351757331369, 6.41347043882812086622548276824, 7.28447967782485622908951424518, 7.63847141098529970365687474861
