| L(s) = 1 | − 2.35·2-s + 3.12·3-s + 3.56·4-s + 5-s − 7.37·6-s + 7-s − 3.68·8-s + 6.76·9-s − 2.35·10-s + 11.1·12-s − 0.303·13-s − 2.35·14-s + 3.12·15-s + 1.56·16-s + 1.03·17-s − 15.9·18-s + 5.12·19-s + 3.56·20-s + 3.12·21-s + 8.52·23-s − 11.5·24-s + 25-s + 0.716·26-s + 11.7·27-s + 3.56·28-s − 4.31·29-s − 7.37·30-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.80·3-s + 1.78·4-s + 0.447·5-s − 3.00·6-s + 0.377·7-s − 1.30·8-s + 2.25·9-s − 0.745·10-s + 3.21·12-s − 0.0842·13-s − 0.630·14-s + 0.807·15-s + 0.392·16-s + 0.250·17-s − 3.76·18-s + 1.17·19-s + 0.796·20-s + 0.682·21-s + 1.77·23-s − 2.35·24-s + 0.200·25-s + 0.140·26-s + 2.26·27-s + 0.673·28-s − 0.802·29-s − 1.34·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237947809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237947809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 3.12T + 3T^{2} \) |
| 13 | \( 1 + 0.303T + 13T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 8.52T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 + 8.81T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 0.573T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−8.615968970899791973122961926338, −7.79715111942751580525400781571, −7.37996994519161848493898393486, −6.84692068675187558311957850467, −5.55517139258232283884399038075, −4.47402607096667868874456379611, −3.28699630723201429325921004080, −2.68605097108593393227753664939, −1.77527845131778071284685655818, −1.10010769324377596596671861121,
1.10010769324377596596671861121, 1.77527845131778071284685655818, 2.68605097108593393227753664939, 3.28699630723201429325921004080, 4.47402607096667868874456379611, 5.55517139258232283884399038075, 6.84692068675187558311957850467, 7.37996994519161848493898393486, 7.79715111942751580525400781571, 8.615968970899791973122961926338
