| L(s) = 1 | − 2.00·3-s + 0.977·5-s − 3.03·7-s + 1.00·9-s + 11-s − 4.12·13-s − 1.95·15-s − 0.328·17-s − 8.36·19-s + 6.07·21-s + 2.85·23-s − 4.04·25-s + 3.98·27-s + 1.72·29-s + 5.02·31-s − 2.00·33-s − 2.96·35-s − 3.87·37-s + 8.26·39-s − 12.5·41-s − 10.5·43-s + 0.986·45-s + 5.81·47-s + 2.20·49-s + 0.658·51-s − 13.3·53-s + 0.977·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.437·5-s − 1.14·7-s + 0.336·9-s + 0.301·11-s − 1.14·13-s − 0.505·15-s − 0.0797·17-s − 1.92·19-s + 1.32·21-s + 0.594·23-s − 0.808·25-s + 0.767·27-s + 0.319·29-s + 0.902·31-s − 0.348·33-s − 0.501·35-s − 0.636·37-s + 1.32·39-s − 1.95·41-s − 1.60·43-s + 0.146·45-s + 0.848·47-s + 0.314·49-s + 0.0922·51-s − 1.82·53-s + 0.131·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3841997525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3841997525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 - 0.977T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 + 0.328T + 17T^{2} \) |
| 19 | \( 1 + 8.36T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.127645579106256459167754659611, −6.91920498377048826230346839238, −6.56070489218595470396536778266, −6.14585612439341518244768798634, −5.18635290568001939504160891434, −4.72197714187067295344299927785, −3.67750199994775621109155314760, −2.74758399245672276939568960928, −1.80908148775847419791691588590, −0.32850942817697334890057945799,
0.32850942817697334890057945799, 1.80908148775847419791691588590, 2.74758399245672276939568960928, 3.67750199994775621109155314760, 4.72197714187067295344299927785, 5.18635290568001939504160891434, 6.14585612439341518244768798634, 6.56070489218595470396536778266, 6.91920498377048826230346839238, 8.127645579106256459167754659611
