| L(s) = 1 | + 3·3-s + 4.63·7-s + 9·9-s + 58.7·11-s − 33.5·13-s − 44.2·17-s − 133.·19-s + 13.9·21-s − 90.0·23-s + 27·27-s + 154.·29-s + 21.0·31-s + 176.·33-s + 38.0·37-s − 100.·39-s + 335.·41-s − 388.·43-s − 267.·47-s − 321.·49-s − 132.·51-s − 445.·53-s − 400.·57-s + 36.6·59-s − 813.·61-s + 41.7·63-s + 41.5·67-s − 270.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.250·7-s + 0.333·9-s + 1.61·11-s − 0.716·13-s − 0.630·17-s − 1.61·19-s + 0.144·21-s − 0.816·23-s + 0.192·27-s + 0.989·29-s + 0.122·31-s + 0.930·33-s + 0.169·37-s − 0.413·39-s + 1.27·41-s − 1.37·43-s − 0.829·47-s − 0.937·49-s − 0.364·51-s − 1.15·53-s − 0.930·57-s + 0.0807·59-s − 1.70·61-s + 0.0834·63-s + 0.0758·67-s − 0.471·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.63T + 343T^{2} \) |
| 11 | \( 1 - 58.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 90.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 21.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 38.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 267.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 445.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 36.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 813.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 41.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 9.99e2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 56.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 80.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 577.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 679.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 193.T + 9.12e5T^{2} \) |
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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
−8.334557868865058079529963567199, −7.55627742077806105227715495838, −6.51853491161420298715920963066, −6.25662965460269013601391321405, −4.69126104854689778342531786203, −4.31377295423595799049801779259, −3.30106552239635956494122244710, −2.21682967288586360785530761200, −1.44722930846223280846137352158, 0,
1.44722930846223280846137352158, 2.21682967288586360785530761200, 3.30106552239635956494122244710, 4.31377295423595799049801779259, 4.69126104854689778342531786203, 6.25662965460269013601391321405, 6.51853491161420298715920963066, 7.55627742077806105227715495838, 8.334557868865058079529963567199
