| L(s) = 1 | + 2.35·2-s + 3.53·4-s + 5-s − 0.0779·7-s + 3.62·8-s + 2.35·10-s + 1.03·11-s + 13-s − 0.183·14-s + 1.44·16-s + 3.01·17-s − 1.33·19-s + 3.53·20-s + 2.44·22-s + 7.32·23-s + 25-s + 2.35·26-s − 0.276·28-s + 4.09·29-s − 6.98·31-s − 3.83·32-s + 7.10·34-s − 0.0779·35-s + 5.18·37-s − 3.15·38-s + 3.62·40-s + 3.31·41-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.76·4-s + 0.447·5-s − 0.0294·7-s + 1.28·8-s + 0.744·10-s + 0.312·11-s + 0.277·13-s − 0.0490·14-s + 0.362·16-s + 0.731·17-s − 0.307·19-s + 0.791·20-s + 0.520·22-s + 1.52·23-s + 0.200·25-s + 0.461·26-s − 0.0521·28-s + 0.760·29-s − 1.25·31-s − 0.678·32-s + 1.21·34-s − 0.0131·35-s + 0.852·37-s − 0.511·38-s + 0.572·40-s + 0.517·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.428471932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.428471932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 7 | \( 1 + 0.0779T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 17 | \( 1 - 3.01T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 23 | \( 1 - 7.32T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 6.98T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 0.0478T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 - 8.38T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 - 1.81T + 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−7.955460753663331042221818158054, −7.15626837948342651922710146040, −6.50051895241128783673535985659, −5.88519447071600060434495927938, −5.24132530450071554180883880441, −4.57877181642578390065052149964, −3.76389214692298660171156645457, −3.05719454741298728486515192153, −2.28031851850374601786047917104, −1.14692237151602387054109367345,
1.14692237151602387054109367345, 2.28031851850374601786047917104, 3.05719454741298728486515192153, 3.76389214692298660171156645457, 4.57877181642578390065052149964, 5.24132530450071554180883880441, 5.88519447071600060434495927938, 6.50051895241128783673535985659, 7.15626837948342651922710146040, 7.955460753663331042221818158054
