| L(s) = 1 | − 1.77·3-s − 0.716·5-s + 0.155·9-s + 1.69·11-s − 4.41·13-s + 1.27·15-s + 1.35·17-s + 6.61·19-s − 4.94·23-s − 4.48·25-s + 5.05·27-s − 0.710·29-s + 3.41·31-s − 3.01·33-s − 5.21·37-s + 7.84·39-s − 41-s − 6.26·43-s − 0.111·45-s + 7.29·47-s − 2.41·51-s − 12.1·53-s − 1.21·55-s − 11.7·57-s − 1.69·59-s + 7.76·61-s + 3.16·65-s + ⋯ |
| L(s) = 1 | − 1.02·3-s − 0.320·5-s + 0.0518·9-s + 0.511·11-s − 1.22·13-s + 0.328·15-s + 0.329·17-s + 1.51·19-s − 1.03·23-s − 0.897·25-s + 0.972·27-s − 0.132·29-s + 0.613·31-s − 0.524·33-s − 0.857·37-s + 1.25·39-s − 0.156·41-s − 0.955·43-s − 0.0166·45-s + 1.06·47-s − 0.337·51-s − 1.67·53-s − 0.164·55-s − 1.55·57-s − 0.220·59-s + 0.994·61-s + 0.392·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7770935169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7770935169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 5 | \( 1 + 0.716T + 5T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 + 0.710T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 1.69T + 59T^{2} \) |
| 61 | \( 1 - 7.76T + 61T^{2} \) |
| 67 | \( 1 - 3.13T + 67T^{2} \) |
| 71 | \( 1 - 3.54T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 6.08T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.82420780953476813831515298285, −7.00479951359984682145767260863, −6.49524098459498868294902924007, −5.54449119051509758132827385046, −5.27227911717776737359047529905, −4.40287351031428937742156067853, −3.59010355443660057502824670648, −2.71336930056397601560812243284, −1.59525522383135092139291866170, −0.46465366664918793496032799846,
0.46465366664918793496032799846, 1.59525522383135092139291866170, 2.71336930056397601560812243284, 3.59010355443660057502824670648, 4.40287351031428937742156067853, 5.27227911717776737359047529905, 5.54449119051509758132827385046, 6.49524098459498868294902924007, 7.00479951359984682145767260863, 7.82420780953476813831515298285
