| L(s) = 1 | + 5·5-s + 24.6·7-s − 62.6·11-s + 59.6·13-s − 85.2·17-s − 75.1·19-s − 97.5·23-s + 25·25-s + 159.·29-s + 318.·31-s + 123.·35-s + 393.·37-s + 85.0·41-s − 40.7·43-s − 90.2·47-s + 262.·49-s + 691.·53-s − 313.·55-s + 283.·59-s − 432.·61-s + 298.·65-s + 308.·67-s + 302.·71-s − 457.·73-s − 1.54e3·77-s − 841.·79-s − 788.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.32·7-s − 1.71·11-s + 1.27·13-s − 1.21·17-s − 0.907·19-s − 0.884·23-s + 0.200·25-s + 1.01·29-s + 1.84·31-s + 0.594·35-s + 1.74·37-s + 0.324·41-s − 0.144·43-s − 0.280·47-s + 0.764·49-s + 1.79·53-s − 0.767·55-s + 0.625·59-s − 0.907·61-s + 0.569·65-s + 0.562·67-s + 0.505·71-s − 0.733·73-s − 2.28·77-s − 1.19·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.582397595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582397595\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + 62.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 318.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 393.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 85.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 40.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 90.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 691.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 283.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 432.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 308.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 457.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 841.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 788.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−8.673241929519805983726147925627, −8.357018276881121317221809844443, −7.66662742693995745891572602196, −6.44729527606126756401853439927, −5.82094477939991391549298683187, −4.78180616296722990436621767447, −4.27433895526948635048474639012, −2.69955269252859919847655559040, −2.02272881094368691081976646121, −0.77635420603034861105399299046,
0.77635420603034861105399299046, 2.02272881094368691081976646121, 2.69955269252859919847655559040, 4.27433895526948635048474639012, 4.78180616296722990436621767447, 5.82094477939991391549298683187, 6.44729527606126756401853439927, 7.66662742693995745891572602196, 8.357018276881121317221809844443, 8.673241929519805983726147925627
