L(s) = 1 | − 3-s − 1.74·7-s + 9-s + 2.32·11-s + 4.93·13-s + 3.74·17-s − 1.69·19-s + 1.74·21-s + 8.67·23-s − 27-s + 3.12·29-s + 9.56·31-s − 2.32·33-s + 5.36·37-s − 4.93·39-s − 8.72·41-s + 2.86·43-s − 8.46·47-s − 3.97·49-s − 3.74·51-s − 1.34·53-s + 1.69·57-s + 4.38·59-s − 15.3·61-s − 1.74·63-s + 9.23·67-s − 8.67·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.657·7-s + 0.333·9-s + 0.699·11-s + 1.36·13-s + 0.907·17-s − 0.388·19-s + 0.379·21-s + 1.80·23-s − 0.192·27-s + 0.579·29-s + 1.71·31-s − 0.403·33-s + 0.882·37-s − 0.790·39-s − 1.36·41-s + 0.436·43-s − 1.23·47-s − 0.567·49-s − 0.523·51-s − 0.184·53-s + 0.224·57-s + 0.571·59-s − 1.96·61-s − 0.219·63-s + 1.12·67-s − 1.04·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.919158225\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.919158225\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 8.67T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 - 4.38T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 9.23T + 67T^{2} \) |
| 71 | \( 1 - 0.0235T + 71T^{2} \) |
| 73 | \( 1 + 1.02T + 73T^{2} \) |
| 79 | \( 1 + 0.798T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 + 4.86T + 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−7.946180123043143153317522218235, −6.91199691019661907040730472331, −6.45940324575830327304484641500, −6.00414896178643412935726890964, −5.06780847740137187752891707260, −4.39256700465692783285167926582, −3.45696532152088931692900934529, −2.94142543635000247546184001749, −1.47277273297439821367278569527, −0.797030282189655593918282481218,
0.797030282189655593918282481218, 1.47277273297439821367278569527, 2.94142543635000247546184001749, 3.45696532152088931692900934529, 4.39256700465692783285167926582, 5.06780847740137187752891707260, 6.00414896178643412935726890964, 6.45940324575830327304484641500, 6.91199691019661907040730472331, 7.946180123043143153317522218235