| L(s) = 1 | + 3-s + 5-s − 1.56·7-s + 9-s + 5.56·11-s + 13-s + 15-s − 6.68·17-s − 3.12·19-s − 1.56·21-s + 5.56·23-s + 25-s + 27-s − 2·29-s + 7.12·31-s + 5.56·33-s − 1.56·35-s + 9.80·37-s + 39-s + 2.68·41-s + 10.2·43-s + 45-s − 7.12·47-s − 4.56·49-s − 6.68·51-s + 3.56·53-s + 5.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.590·7-s + 0.333·9-s + 1.67·11-s + 0.277·13-s + 0.258·15-s − 1.62·17-s − 0.716·19-s − 0.340·21-s + 1.15·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s + 1.27·31-s + 0.968·33-s − 0.263·35-s + 1.61·37-s + 0.160·39-s + 0.419·41-s + 1.56·43-s + 0.149·45-s − 1.03·47-s − 0.651·49-s − 0.936·51-s + 0.489·53-s + 0.749·55-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
| \(L(1)\) |
\(\approx\) |
\(2.625294796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625294796\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 - 3.56T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.815014694413489598571412062105, −8.148571135651191144328245380765, −6.86779272285683299483463009337, −6.63984414580381239027607375686, −5.87947871766561781153467074195, −4.52457942525419756444093845293, −4.06720981588091634387352327185, −2.99290795806884686627078635784, −2.14692916635221182113908744821, −0.997917899636925866747111329554,
0.997917899636925866747111329554, 2.14692916635221182113908744821, 2.99290795806884686627078635784, 4.06720981588091634387352327185, 4.52457942525419756444093845293, 5.87947871766561781153467074195, 6.63984414580381239027607375686, 6.86779272285683299483463009337, 8.148571135651191144328245380765, 8.815014694413489598571412062105
