L(s) = 1 | − 2.26·3-s − 0.762·5-s − 1.25·7-s + 2.12·9-s + 1.90·11-s − 2.67·13-s + 1.72·15-s + 7.34·17-s + 6.08·19-s + 2.84·21-s + 0.363·23-s − 4.41·25-s + 1.98·27-s + 8.19·29-s + 5.80·31-s − 4.31·33-s + 0.957·35-s − 6.69·37-s + 6.05·39-s − 5.03·41-s + 8.74·43-s − 1.61·45-s − 7.61·47-s − 5.42·49-s − 16.6·51-s − 3.07·53-s − 1.45·55-s + ⋯ |
L(s) = 1 | − 1.30·3-s − 0.341·5-s − 0.474·7-s + 0.707·9-s + 0.575·11-s − 0.741·13-s + 0.445·15-s + 1.78·17-s + 1.39·19-s + 0.620·21-s + 0.0758·23-s − 0.883·25-s + 0.381·27-s + 1.52·29-s + 1.04·31-s − 0.751·33-s + 0.161·35-s − 1.09·37-s + 0.969·39-s − 0.785·41-s + 1.33·43-s − 0.241·45-s − 1.11·47-s − 0.774·49-s − 2.32·51-s − 0.422·53-s − 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038330229\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038330229\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 + 0.762T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 1.90T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 - 0.363T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 6.69T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 + 3.36T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 7.16T + 73T^{2} \) |
| 79 | \( 1 + 9.28T + 79T^{2} \) |
| 83 | \( 1 - 1.97T + 83T^{2} \) |
| 89 | \( 1 + 8.78T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.76490375246265025772167456618, −6.92967792450120349648791020425, −6.49049540266672135221427005351, −5.63253513211305275595874344447, −5.22262134776652016022625024880, −4.47510232090731024223928565335, −3.48673395043885879876925391150, −2.87703460256411403751064640267, −1.38161045759492219915839005529, −0.59522084059715159166516271564,
0.59522084059715159166516271564, 1.38161045759492219915839005529, 2.87703460256411403751064640267, 3.48673395043885879876925391150, 4.47510232090731024223928565335, 5.22262134776652016022625024880, 5.63253513211305275595874344447, 6.49049540266672135221427005351, 6.92967792450120349648791020425, 7.76490375246265025772167456618