| L(s) = 1 | + 2-s − 1.11·3-s + 4-s + 1.58·5-s − 1.11·6-s + 8-s − 1.75·9-s + 1.58·10-s − 0.769·11-s − 1.11·12-s − 3.30·13-s − 1.76·15-s + 16-s − 0.964·17-s − 1.75·18-s + 4.13·19-s + 1.58·20-s − 0.769·22-s − 8.05·23-s − 1.11·24-s − 2.49·25-s − 3.30·26-s + 5.30·27-s + 4.60·29-s − 1.76·30-s + 4.73·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.644·3-s + 0.5·4-s + 0.708·5-s − 0.455·6-s + 0.353·8-s − 0.584·9-s + 0.500·10-s − 0.231·11-s − 0.322·12-s − 0.916·13-s − 0.456·15-s + 0.250·16-s − 0.233·17-s − 0.413·18-s + 0.948·19-s + 0.354·20-s − 0.163·22-s − 1.68·23-s − 0.227·24-s − 0.498·25-s − 0.647·26-s + 1.02·27-s + 0.854·29-s − 0.323·30-s + 0.850·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.394229116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394229116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 + 0.769T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 0.964T + 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 + 8.05T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 - 8.62T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 - 0.840T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 8.13T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 1.80T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.069890199930117876551568937371, −7.47971442686565268677934569240, −6.53581543607733711818102889217, −5.91111149518387227354943260374, −5.49654450323140205799064225364, −4.71515779420463570941967810674, −3.92276651989936768098732246746, −2.70039236428656555793568320832, −2.24203238391682194074713407125, −0.77277908695560679265506202511,
0.77277908695560679265506202511, 2.24203238391682194074713407125, 2.70039236428656555793568320832, 3.92276651989936768098732246746, 4.71515779420463570941967810674, 5.49654450323140205799064225364, 5.91111149518387227354943260374, 6.53581543607733711818102889217, 7.47971442686565268677934569240, 8.069890199930117876551568937371
