| L(s) = 1 | − 3·3-s − 17.8·5-s + 9·9-s + 11.3·11-s + 13.0·13-s + 53.6·15-s − 53.2·17-s − 42.4·19-s − 152.·23-s + 194.·25-s − 27·27-s + 186.·29-s − 157.·31-s − 34.1·33-s + 3.74·37-s − 39.2·39-s + 39.3·41-s − 429.·43-s − 160.·45-s + 21.1·47-s + 159.·51-s + 365.·53-s − 203.·55-s + 127.·57-s − 226.·59-s − 651.·61-s − 234.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.59·5-s + 0.333·9-s + 0.312·11-s + 0.279·13-s + 0.922·15-s − 0.759·17-s − 0.512·19-s − 1.37·23-s + 1.55·25-s − 0.192·27-s + 1.19·29-s − 0.914·31-s − 0.180·33-s + 0.0166·37-s − 0.161·39-s + 0.149·41-s − 1.52·43-s − 0.532·45-s + 0.0657·47-s + 0.438·51-s + 0.948·53-s − 0.499·55-s + 0.295·57-s − 0.499·59-s − 1.36·61-s − 0.446·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4404485549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4404485549\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 17.8T + 125T^{2} \) |
| 11 | \( 1 - 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 186.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 3.74T + 5.06e4T^{2} \) |
| 41 | \( 1 - 39.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 21.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 365.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 651.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 368.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 608.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 910.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 327.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 37.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 722.T + 9.12e5T^{2} \) |
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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
−8.423316468827777209821294499191, −7.954711290494887036828830607837, −7.01674123585200510426189797005, −6.47267960020091642920698383180, −5.46616555447345917113362692692, −4.33658875440236832786156947264, −4.08763739685224521908835903157, −2.99116409375745556868359269578, −1.60342633881506584407655446662, −0.30691012391399074173396919439,
0.30691012391399074173396919439, 1.60342633881506584407655446662, 2.99116409375745556868359269578, 4.08763739685224521908835903157, 4.33658875440236832786156947264, 5.46616555447345917113362692692, 6.47267960020091642920698383180, 7.01674123585200510426189797005, 7.954711290494887036828830607837, 8.423316468827777209821294499191
