| L(s) = 1 | − 3.51·5-s − 7-s − 0.779·11-s − 1.36·13-s − 6.75·17-s − 19-s + 5.71·23-s + 7.32·25-s − 4.29·29-s + 4.32·31-s + 3.51·35-s − 6.65·37-s − 8.70·41-s − 4.32·43-s − 7.66·47-s + 49-s − 7.53·53-s + 2.73·55-s − 6.65·61-s + 4.80·65-s − 0.960·67-s + 5.97·71-s + 11.9·73-s + 0.779·77-s + 8.96·79-s + 3.88·83-s + 23.7·85-s + ⋯ |
| L(s) = 1 | − 1.57·5-s − 0.377·7-s − 0.234·11-s − 0.379·13-s − 1.63·17-s − 0.229·19-s + 1.19·23-s + 1.46·25-s − 0.796·29-s + 0.777·31-s + 0.593·35-s − 1.09·37-s − 1.36·41-s − 0.660·43-s − 1.11·47-s + 0.142·49-s − 1.03·53-s + 0.368·55-s − 0.852·61-s + 0.595·65-s − 0.117·67-s + 0.709·71-s + 1.39·73-s + 0.0887·77-s + 1.00·79-s + 0.426·83-s + 2.57·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5718877744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5718877744\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 + 0.779T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 - 4.32T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.65T + 61T^{2} \) |
| 67 | \( 1 + 0.960T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 8.96T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 3.77T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.299059061160017945646896691509, −7.55380476862569822671766122416, −6.87297861033597097714864088735, −6.40449450896473975923026910310, −4.99765724387899710884929236086, −4.67118412439856927074724404387, −3.65743047197526392009454039906, −3.12178518949581915028880520518, −1.96923964101389673472215954539, −0.39708426359086448397722834034,
0.39708426359086448397722834034, 1.96923964101389673472215954539, 3.12178518949581915028880520518, 3.65743047197526392009454039906, 4.67118412439856927074724404387, 4.99765724387899710884929236086, 6.40449450896473975923026910310, 6.87297861033597097714864088735, 7.55380476862569822671766122416, 8.299059061160017945646896691509
