| L(s) = 1 | + 1.79·5-s + 7-s − 2.96·11-s + 6.95·13-s − 6.11·17-s − 0.967·19-s − 7.91·23-s − 1.76·25-s + 0.798·29-s − 0.798·31-s + 1.79·35-s + 6.76·37-s + 3.79·41-s + 9.71·43-s + 11.0·47-s + 49-s + 13.6·53-s − 5.33·55-s − 4.59·59-s + 8.56·61-s + 12.4·65-s − 13.5·67-s + 11.9·71-s + 4.33·73-s − 2.96·77-s + 17.6·79-s + 0.765·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s + 0.377·7-s − 0.894·11-s + 1.92·13-s − 1.48·17-s − 0.221·19-s − 1.65·23-s − 0.353·25-s + 0.148·29-s − 0.143·31-s + 0.303·35-s + 1.11·37-s + 0.593·41-s + 1.48·43-s + 1.61·47-s + 0.142·49-s + 1.87·53-s − 0.719·55-s − 0.598·59-s + 1.09·61-s + 1.55·65-s − 1.65·67-s + 1.41·71-s + 0.507·73-s − 0.338·77-s + 1.98·79-s + 0.0840·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.374500174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.374500174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 - 6.95T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 + 0.967T + 19T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 - 0.798T + 29T^{2} \) |
| 31 | \( 1 + 0.798T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 4.33T + 73T^{2} \) |
| 79 | \( 1 - 17.6T + 79T^{2} \) |
| 83 | \( 1 - 0.765T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 + 6.86T + 97T^{2} \) |
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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
−8.108992350154335867091639720837, −7.48501771554686146287935688905, −6.37109010317900031637652177016, −6.01837818957401838296908463620, −5.40412128366557831389440500139, −4.28077577935021345627136185319, −3.85712473997849002561051989550, −2.46355349101495931165314644410, −2.04253846886696217174063333670, −0.810036351205887702786946299486,
0.810036351205887702786946299486, 2.04253846886696217174063333670, 2.46355349101495931165314644410, 3.85712473997849002561051989550, 4.28077577935021345627136185319, 5.40412128366557831389440500139, 6.01837818957401838296908463620, 6.37109010317900031637652177016, 7.48501771554686146287935688905, 8.108992350154335867091639720837
