| L(s) = 1 | − 1.26·2-s + 0.320·3-s − 0.404·4-s − 0.404·6-s − 7-s + 3.03·8-s − 2.89·9-s − 5.62·11-s − 0.129·12-s + 6.38·13-s + 1.26·14-s − 3.02·16-s + 1.45·17-s + 3.65·18-s + 8.36·19-s − 0.320·21-s + 7.10·22-s + 23-s + 0.973·24-s − 8.06·26-s − 1.88·27-s + 0.404·28-s − 2.94·29-s + 0.597·31-s − 2.25·32-s − 1.80·33-s − 1.83·34-s + ⋯ |
| L(s) = 1 | − 0.893·2-s + 0.184·3-s − 0.202·4-s − 0.165·6-s − 0.377·7-s + 1.07·8-s − 0.965·9-s − 1.69·11-s − 0.0374·12-s + 1.77·13-s + 0.337·14-s − 0.756·16-s + 0.352·17-s + 0.862·18-s + 1.91·19-s − 0.0699·21-s + 1.51·22-s + 0.208·23-s + 0.198·24-s − 1.58·26-s − 0.363·27-s + 0.0764·28-s − 0.546·29-s + 0.107·31-s − 0.397·32-s − 0.313·33-s − 0.314·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7698353602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7698353602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 - 0.320T + 3T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 - 6.38T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 - 8.36T + 19T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 - 0.597T + 31T^{2} \) |
| 37 | \( 1 - 0.962T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 6.05T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 0.178T + 61T^{2} \) |
| 67 | \( 1 - 9.63T + 67T^{2} \) |
| 71 | \( 1 + 1.23T + 71T^{2} \) |
| 73 | \( 1 - 4.30T + 73T^{2} \) |
| 79 | \( 1 - 0.418T + 79T^{2} \) |
| 83 | \( 1 + 3.61T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 + 8.51T + 97T^{2} \) |
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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
−8.425017741030742485956630463394, −7.935057324215953245613991692699, −7.35628942267534520628312488291, −6.21750195273028198820146812407, −5.45273668078405780363701538310, −4.89139244983978672999542022623, −3.47261800953766449724878802678, −3.09734077288245370637433575226, −1.71994142238914331478894842549, −0.57829733446708080708220468562,
0.57829733446708080708220468562, 1.71994142238914331478894842549, 3.09734077288245370637433575226, 3.47261800953766449724878802678, 4.89139244983978672999542022623, 5.45273668078405780363701538310, 6.21750195273028198820146812407, 7.35628942267534520628312488291, 7.935057324215953245613991692699, 8.425017741030742485956630463394
