| L(s) = 1 | + 2·2-s + 8.38·3-s + 4·4-s + 16.7·6-s + 23.6·7-s + 8·8-s + 43.3·9-s + 70.9·11-s + 33.5·12-s − 30.9·13-s + 47.3·14-s + 16·16-s − 88.5·17-s + 86.7·18-s + 25.5·19-s + 198.·21-s + 141.·22-s − 23·23-s + 67.1·24-s − 61.8·26-s + 137.·27-s + 94.7·28-s − 219.·29-s + 260.·31-s + 32·32-s + 595.·33-s − 177.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.61·3-s + 0.5·4-s + 1.14·6-s + 1.27·7-s + 0.353·8-s + 1.60·9-s + 1.94·11-s + 0.807·12-s − 0.660·13-s + 0.904·14-s + 0.250·16-s − 1.26·17-s + 1.13·18-s + 0.308·19-s + 2.06·21-s + 1.37·22-s − 0.208·23-s + 0.570·24-s − 0.466·26-s + 0.980·27-s + 0.639·28-s − 1.40·29-s + 1.51·31-s + 0.176·32-s + 3.14·33-s − 0.893·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.769202616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.769202616\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 8.38T + 27T^{2} \) |
| 7 | \( 1 - 23.6T + 343T^{2} \) |
| 11 | \( 1 - 70.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.5T + 6.85e3T^{2} \) |
| 29 | \( 1 + 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 50.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 261.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 403.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 475.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 263.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 425.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 238.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 23.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 634.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 163.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 293.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 792.T + 9.12e5T^{2} \) |
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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
−9.140442257951794737356593238339, −8.700614441884460979202321596552, −7.74457372458479016674607123106, −7.11782224316698279721878739046, −6.12125629489767293388815943432, −4.64753793261395354049289963774, −4.22475912126828158822899497961, −3.23140583024452000320869801799, −2.11845854395056563521773519704, −1.46681024420838391214733162662,
1.46681024420838391214733162662, 2.11845854395056563521773519704, 3.23140583024452000320869801799, 4.22475912126828158822899497961, 4.64753793261395354049289963774, 6.12125629489767293388815943432, 7.11782224316698279721878739046, 7.74457372458479016674607123106, 8.700614441884460979202321596552, 9.140442257951794737356593238339
