| L(s) = 1 | + 2-s + 3.35·3-s + 4-s − 1.35·5-s + 3.35·6-s − 7-s + 8-s + 8.28·9-s − 1.35·10-s + 0.672·11-s + 3.35·12-s − 1.04·13-s − 14-s − 4.56·15-s + 16-s − 2.46·17-s + 8.28·18-s − 1.35·20-s − 3.35·21-s + 0.672·22-s + 9.22·23-s + 3.35·24-s − 3.15·25-s − 1.04·26-s + 17.7·27-s − 28-s + 9.26·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.93·3-s + 0.5·4-s − 0.607·5-s + 1.37·6-s − 0.377·7-s + 0.353·8-s + 2.76·9-s − 0.429·10-s + 0.202·11-s + 0.969·12-s − 0.289·13-s − 0.267·14-s − 1.17·15-s + 0.250·16-s − 0.597·17-s + 1.95·18-s − 0.303·20-s − 0.733·21-s + 0.143·22-s + 1.92·23-s + 0.685·24-s − 0.630·25-s − 0.204·26-s + 3.41·27-s − 0.188·28-s + 1.71·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.094548934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.094548934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.35T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 11 | \( 1 - 0.672T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 23 | \( 1 - 9.22T + 23T^{2} \) |
| 29 | \( 1 - 9.26T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 37 | \( 1 + 6.98T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 - 0.0812T + 43T^{2} \) |
| 47 | \( 1 + 0.226T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.07T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.358T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 9.58T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
| 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
−8.102750714564560424903298786428, −7.64004859986493469306433548371, −6.87596806504709704336425689723, −6.36562876145746562652464121599, −4.77286249800679859876872966946, −4.49572542615272239283163932072, −3.48867785592124576821965369184, −3.03495781863020434800185340628, −2.34468457307457169713052081355, −1.19955149491777920860654790241,
1.19955149491777920860654790241, 2.34468457307457169713052081355, 3.03495781863020434800185340628, 3.48867785592124576821965369184, 4.49572542615272239283163932072, 4.77286249800679859876872966946, 6.36562876145746562652464121599, 6.87596806504709704336425689723, 7.64004859986493469306433548371, 8.102750714564560424903298786428
