L(s) = 1 | − 3-s − 5-s − 2.61·7-s + 9-s + 6.39·11-s + 13-s + 15-s − 0.615·17-s − 7.77·19-s + 2.61·21-s − 2.61·23-s + 25-s − 27-s + 7.00·29-s − 5.23·31-s − 6.39·33-s + 2.61·35-s + 7.16·37-s − 39-s + 7.16·41-s − 4·43-s − 45-s + 1.23·47-s − 0.161·49-s + 0.615·51-s − 13.6·53-s − 6.39·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.988·7-s + 0.333·9-s + 1.92·11-s + 0.277·13-s + 0.258·15-s − 0.149·17-s − 1.78·19-s + 0.570·21-s − 0.545·23-s + 0.200·25-s − 0.192·27-s + 1.30·29-s − 0.939·31-s − 1.11·33-s + 0.442·35-s + 1.17·37-s − 0.160·39-s + 1.11·41-s − 0.609·43-s − 0.149·45-s + 0.179·47-s − 0.0230·49-s + 0.0861·51-s − 1.87·53-s − 0.861·55-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
\(L(1)\) |
\(\approx\) |
\(1.129951639\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.129951639\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 17 | \( 1 + 0.615T + 17T^{2} \) |
| 19 | \( 1 + 7.77T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 0.391T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 - 0.993T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
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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
−8.792720284769265885999232785468, −7.970047730464109918046349024333, −6.88001119839502555977272393857, −6.39370396633059569680846969807, −6.02591709384105128700720491627, −4.59018588183955064901716931725, −4.07202580031588541264983123107, −3.28655186559322266382989149359, −1.93083545985139733897075638198, −0.66172664170434032537120859125,
0.66172664170434032537120859125, 1.93083545985139733897075638198, 3.28655186559322266382989149359, 4.07202580031588541264983123107, 4.59018588183955064901716931725, 6.02591709384105128700720491627, 6.39370396633059569680846969807, 6.88001119839502555977272393857, 7.970047730464109918046349024333, 8.792720284769265885999232785468