| L(s) = 1 | + 2-s + 4-s − 3.23·5-s + 8-s − 3.23·10-s − 1.85·11-s − 4.85·13-s + 16-s − 6.47·17-s + 3.61·19-s − 3.23·20-s − 1.85·22-s − 2·23-s + 5.47·25-s − 4.85·26-s + 7.09·29-s + 6·31-s + 32-s − 6.47·34-s + 6.47·37-s + 3.61·38-s − 3.23·40-s + 10·41-s + 8.61·43-s − 1.85·44-s − 2·46-s − 3.38·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.44·5-s + 0.353·8-s − 1.02·10-s − 0.559·11-s − 1.34·13-s + 0.250·16-s − 1.56·17-s + 0.830·19-s − 0.723·20-s − 0.395·22-s − 0.417·23-s + 1.09·25-s − 0.951·26-s + 1.31·29-s + 1.07·31-s + 0.176·32-s − 1.10·34-s + 1.06·37-s + 0.586·38-s − 0.511·40-s + 1.56·41-s + 1.31·43-s − 0.279·44-s − 0.294·46-s − 0.493·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.660875897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660875897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 8.61T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 - 8.09T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.998880815912684277848130424573, −7.83501999467160431049448764484, −6.97441107178836440461893221581, −6.32918184743024021744170077376, −5.20270588356444709703977765050, −4.51392404481854218452648312241, −4.11596438084843266142720894638, −2.93143559232357832146625557390, −2.40726286259716874662643794368, −0.63450177945590910051913315988,
0.63450177945590910051913315988, 2.40726286259716874662643794368, 2.93143559232357832146625557390, 4.11596438084843266142720894638, 4.51392404481854218452648312241, 5.20270588356444709703977765050, 6.32918184743024021744170077376, 6.97441107178836440461893221581, 7.83501999467160431049448764484, 7.998880815912684277848130424573
