| L(s) = 1 | + 5·2-s + 2·3-s + 17·4-s − 5·5-s + 10·6-s − 12·7-s + 45·8-s − 23·9-s − 25·10-s + 14·11-s + 34·12-s − 13·13-s − 60·14-s − 10·15-s + 89·16-s + 98·17-s − 115·18-s − 26·19-s − 85·20-s − 24·21-s + 70·22-s − 114·23-s + 90·24-s + 25·25-s − 65·26-s − 100·27-s − 204·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.384·3-s + 17/8·4-s − 0.447·5-s + 0.680·6-s − 0.647·7-s + 1.98·8-s − 0.851·9-s − 0.790·10-s + 0.383·11-s + 0.817·12-s − 0.277·13-s − 1.14·14-s − 0.172·15-s + 1.39·16-s + 1.39·17-s − 1.50·18-s − 0.313·19-s − 0.950·20-s − 0.249·21-s + 0.678·22-s − 1.03·23-s + 0.765·24-s + 1/5·25-s − 0.490·26-s − 0.712·27-s − 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.499735426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.499735426\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 98 T + p^{3} T^{2} \) |
| 19 | \( 1 + 26 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 306 T + p^{3} T^{2} \) |
| 37 | \( 1 - 86 T + p^{3} T^{2} \) |
| 41 | \( 1 + 374 T + p^{3} T^{2} \) |
| 43 | \( 1 + 314 T + p^{3} T^{2} \) |
| 47 | \( 1 - 620 T + p^{3} T^{2} \) |
| 53 | \( 1 - 362 T + p^{3} T^{2} \) |
| 59 | \( 1 - 266 T + p^{3} T^{2} \) |
| 61 | \( 1 - 634 T + p^{3} T^{2} \) |
| 67 | \( 1 - 612 T + p^{3} T^{2} \) |
| 71 | \( 1 + 686 T + p^{3} T^{2} \) |
| 73 | \( 1 - 202 T + p^{3} T^{2} \) |
| 79 | \( 1 + 516 T + p^{3} T^{2} \) |
| 83 | \( 1 - 48 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1230 T + p^{3} T^{2} \) |
| 97 | \( 1 - 350 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27790623736362198666745566235, −13.48526821772808901128583364917, −12.21422099994831154417503430207, −11.69613893361907755942731042642, −10.07920031905832709193227944301, −8.213403723075941243405850690839, −6.69524362340897852949903171040, −5.51849607478230877729023082150, −3.93167111419378412469067164348, −2.81846765206988858819254034886,
2.81846765206988858819254034886, 3.93167111419378412469067164348, 5.51849607478230877729023082150, 6.69524362340897852949903171040, 8.213403723075941243405850690839, 10.07920031905832709193227944301, 11.69613893361907755942731042642, 12.21422099994831154417503430207, 13.48526821772808901128583364917, 14.27790623736362198666745566235
