L(s) = 1 | − 3.21·3-s + 2.30·5-s + 1.34·7-s + 7.31·9-s − 11-s − 1.34·13-s − 7.41·15-s + 0.904·17-s − 19-s − 4.31·21-s − 0.548·23-s + 0.324·25-s − 13.8·27-s − 4.32·29-s − 1.89·31-s + 3.21·33-s + 3.10·35-s − 0.546·37-s + 4.31·39-s − 4.86·41-s − 5.24·43-s + 16.8·45-s + 5.11·47-s − 5.19·49-s − 2.90·51-s − 3.63·53-s − 2.30·55-s + ⋯ |
L(s) = 1 | − 1.85·3-s + 1.03·5-s + 0.508·7-s + 2.43·9-s − 0.301·11-s − 0.373·13-s − 1.91·15-s + 0.219·17-s − 0.229·19-s − 0.942·21-s − 0.114·23-s + 0.0648·25-s − 2.66·27-s − 0.803·29-s − 0.340·31-s + 0.559·33-s + 0.524·35-s − 0.0898·37-s + 0.691·39-s − 0.759·41-s − 0.799·43-s + 2.51·45-s + 0.745·47-s − 0.741·49-s − 0.406·51-s − 0.499·53-s − 0.311·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 - 0.904T + 17T^{2} \) |
| 23 | \( 1 + 0.548T + 23T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 + 0.546T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 5.11T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 - 1.28T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 7.06T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.071414982928944551457707870978, −7.30291018359769037770459087723, −6.47279518295644259323036065949, −5.97524352492748728477794927121, −5.16563313404576267716513931810, −4.89738939191696392152618624346, −3.73743409446342192080737076317, −2.15586229890968112127383941388, −1.36049172356529115273911667401, 0,
1.36049172356529115273911667401, 2.15586229890968112127383941388, 3.73743409446342192080737076317, 4.89738939191696392152618624346, 5.16563313404576267716513931810, 5.97524352492748728477794927121, 6.47279518295644259323036065949, 7.30291018359769037770459087723, 8.071414982928944551457707870978