| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 35.4·7-s + 8·8-s + 10·10-s + 16.6·11-s − 79.9·13-s − 70.8·14-s + 16·16-s + 46.8·17-s − 110.·19-s + 20·20-s + 33.2·22-s + 23·23-s + 25·25-s − 159.·26-s − 141.·28-s + 0.836·29-s − 119.·31-s + 32·32-s + 93.6·34-s − 177.·35-s + 368.·37-s − 221.·38-s + 40·40-s + 95.7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.91·7-s + 0.353·8-s + 0.316·10-s + 0.455·11-s − 1.70·13-s − 1.35·14-s + 0.250·16-s + 0.667·17-s − 1.33·19-s + 0.223·20-s + 0.322·22-s + 0.208·23-s + 0.200·25-s − 1.20·26-s − 0.956·28-s + 0.00535·29-s − 0.694·31-s + 0.176·32-s + 0.472·34-s − 0.855·35-s + 1.63·37-s − 0.944·38-s + 0.158·40-s + 0.364·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.333141510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.333141510\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 + 35.4T + 343T^{2} \) |
| 11 | \( 1 - 16.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 0.836T + 2.43e4T^{2} \) |
| 31 | \( 1 + 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 368.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 331.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 535.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 91.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 329.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 753.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 271.T + 9.12e5T^{2} \) |
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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
−9.124105540437047103299145200312, −7.67282084948072386443599003765, −7.03488983972451791985354318786, −6.23629001997672875113452356278, −5.79399871380085064876019243066, −4.66025198416688089646774880672, −3.82820270861479506528859621878, −2.86229806421714376891858656027, −2.24959710240942508583289448674, −0.59165253085540314681533273102,
0.59165253085540314681533273102, 2.24959710240942508583289448674, 2.86229806421714376891858656027, 3.82820270861479506528859621878, 4.66025198416688089646774880672, 5.79399871380085064876019243066, 6.23629001997672875113452356278, 7.03488983972451791985354318786, 7.67282084948072386443599003765, 9.124105540437047103299145200312
