| L(s) = 1 | + 4.46·5-s − 3·9-s + 5.92·17-s + 14.9·25-s + 1.53·29-s + 11.3·37-s − 11.9·41-s − 13.3·45-s − 7·49-s − 3.53·53-s + 15.3·61-s − 10.8·73-s + 9·81-s + 26.4·85-s + 10·89-s + 18·97-s + 18.3·101-s + 6·109-s − 6.85·113-s + ⋯ |
| L(s) = 1 | + 1.99·5-s − 9-s + 1.43·17-s + 2.98·25-s + 0.285·29-s + 1.87·37-s − 1.86·41-s − 1.99·45-s − 49-s − 0.485·53-s + 1.97·61-s − 1.27·73-s + 81-s + 2.87·85-s + 1.05·89-s + 1.82·97-s + 1.82·101-s + 0.574·109-s − 0.644·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.998376184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.998376184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 4.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.339404407806784979805063868463, −7.44269096659002166816521247633, −6.39600320238416325869614619073, −6.08506653508983395876665948111, −5.34041642192029349053559932620, −4.87616598719280955289602771171, −3.41617768225560963040881789265, −2.74152690375811162388719710257, −1.93205456336423180723752762190, −0.969544949121715982708019921159,
0.969544949121715982708019921159, 1.93205456336423180723752762190, 2.74152690375811162388719710257, 3.41617768225560963040881789265, 4.87616598719280955289602771171, 5.34041642192029349053559932620, 6.08506653508983395876665948111, 6.39600320238416325869614619073, 7.44269096659002166816521247633, 8.339404407806784979805063868463
