L(s) = 1 | − 3-s + 5-s − 3.56·7-s + 9-s − 3.56·11-s − 13-s − 15-s + 0.438·17-s + 8.24·19-s + 3.56·21-s + 4.68·23-s + 25-s − 27-s + 8.24·29-s + 2·31-s + 3.56·33-s − 3.56·35-s − 3.56·37-s + 39-s + 0.438·41-s − 6.24·43-s + 45-s − 10·47-s + 5.68·49-s − 0.438·51-s − 7.56·53-s − 3.56·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.34·7-s + 0.333·9-s − 1.07·11-s − 0.277·13-s − 0.258·15-s + 0.106·17-s + 1.89·19-s + 0.777·21-s + 0.976·23-s + 0.200·25-s − 0.192·27-s + 1.53·29-s + 0.359·31-s + 0.619·33-s − 0.602·35-s − 0.585·37-s + 0.160·39-s + 0.0684·41-s − 0.952·43-s + 0.149·45-s − 1.45·47-s + 0.812·49-s − 0.0613·51-s − 1.03·53-s − 0.480·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 - 8.24T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 0.438T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 7.80T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−8.295459862741191836366426716587, −7.38335680708902944724423216033, −6.76098437633260733894448116856, −6.05146883656682760941857500636, −5.26261330962145684771709161699, −4.70261506558385583493030752457, −3.16822961883214977475307276114, −2.91330545058928577880743304605, −1.30747236893188544551075296630, 0,
1.30747236893188544551075296630, 2.91330545058928577880743304605, 3.16822961883214977475307276114, 4.70261506558385583493030752457, 5.26261330962145684771709161699, 6.05146883656682760941857500636, 6.76098437633260733894448116856, 7.38335680708902944724423216033, 8.295459862741191836366426716587