| L(s) = 1 | − 1.86·3-s − 5-s − 1.14·7-s + 0.466·9-s + 2.22·11-s − 1.50·13-s + 1.86·15-s + 6.80·17-s + 2.20·19-s + 2.12·21-s − 6.80·23-s + 25-s + 4.71·27-s − 7.35·29-s + 0.235·31-s − 4.15·33-s + 1.14·35-s − 1.37·37-s + 2.79·39-s + 5.07·41-s − 0.871·43-s − 0.466·45-s + 6.95·47-s − 5.69·49-s − 12.6·51-s + 3.03·53-s − 2.22·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s − 0.447·5-s − 0.431·7-s + 0.155·9-s + 0.672·11-s − 0.416·13-s + 0.480·15-s + 1.65·17-s + 0.505·19-s + 0.464·21-s − 1.41·23-s + 0.200·25-s + 0.907·27-s − 1.36·29-s + 0.0423·31-s − 0.722·33-s + 0.193·35-s − 0.226·37-s + 0.448·39-s + 0.792·41-s − 0.132·43-s − 0.0695·45-s + 1.01·47-s − 0.813·49-s − 1.77·51-s + 0.416·53-s − 0.300·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8136800038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8136800038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + 1.86T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + 1.50T + 13T^{2} \) |
| 17 | \( 1 - 6.80T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 + 6.80T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 - 0.235T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 + 0.871T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 - 3.03T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 0.145T + 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 0.306T + 73T^{2} \) |
| 79 | \( 1 + 7.61T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.78621947719768315846513183598, −7.45225358998657635690506409796, −6.51051643481408304420665681958, −5.84936579346067418561562229082, −5.44323742951039420165850524124, −4.47637663345040945104513944712, −3.70442471927632215986564222829, −2.95047464344017193445648280456, −1.60189607823745505463921260934, −0.51203676270265928158894960952,
0.51203676270265928158894960952, 1.60189607823745505463921260934, 2.95047464344017193445648280456, 3.70442471927632215986564222829, 4.47637663345040945104513944712, 5.44323742951039420165850524124, 5.84936579346067418561562229082, 6.51051643481408304420665681958, 7.45225358998657635690506409796, 7.78621947719768315846513183598
