L(s) = 1 | + 3.41·3-s + 5-s + 2·7-s + 8.65·9-s − 4.24·11-s − 13-s + 3.41·15-s − 4.82·17-s + 8.24·19-s + 6.82·21-s − 5.07·23-s + 25-s + 19.3·27-s − 9.65·29-s + 1.41·31-s − 14.4·33-s + 2·35-s − 1.17·37-s − 3.41·39-s − 0.828·41-s + 1.75·43-s + 8.65·45-s − 2·47-s − 3·49-s − 16.4·51-s − 3.17·53-s − 4.24·55-s + ⋯ |
L(s) = 1 | + 1.97·3-s + 0.447·5-s + 0.755·7-s + 2.88·9-s − 1.27·11-s − 0.277·13-s + 0.881·15-s − 1.17·17-s + 1.89·19-s + 1.49·21-s − 1.05·23-s + 0.200·25-s + 3.71·27-s − 1.79·29-s + 0.254·31-s − 2.52·33-s + 0.338·35-s − 0.192·37-s − 0.546·39-s − 0.129·41-s + 0.267·43-s + 1.29·45-s − 0.291·47-s − 0.428·49-s − 2.30·51-s − 0.435·53-s − 0.572·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.477772502\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.477772502\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 8.24T + 19T^{2} \) |
| 23 | \( 1 + 5.07T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 - 5.41T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 + 0.828T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−9.662622819620230182144732759631, −9.148036479734114390992896143504, −8.125147413604882015254561565156, −7.76469311471375916617867788197, −6.96898348961327646163272204118, −5.42623865756836789196905133741, −4.52190644736172619743490753065, −3.42415893448044479534166445825, −2.46617969919046215869151829195, −1.72500167905321075740144568774,
1.72500167905321075740144568774, 2.46617969919046215869151829195, 3.42415893448044479534166445825, 4.52190644736172619743490753065, 5.42623865756836789196905133741, 6.96898348961327646163272204118, 7.76469311471375916617867788197, 8.125147413604882015254561565156, 9.148036479734114390992896143504, 9.662622819620230182144732759631