| L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s − 2·12-s + 2·13-s + 4·16-s + 5·19-s + 21-s + 23-s + 27-s − 2·28-s − 8·29-s − 9·31-s − 2·36-s + 9·37-s + 2·39-s + 12·41-s + 4·43-s − 10·47-s + 4·48-s − 6·49-s − 4·52-s + 5·57-s + 12·59-s − 61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 0.554·13-s + 16-s + 1.14·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s − 0.377·28-s − 1.48·29-s − 1.61·31-s − 1/3·36-s + 1.47·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s − 1.45·47-s + 0.577·48-s − 6/7·49-s − 0.554·52-s + 0.662·57-s + 1.56·59-s − 0.128·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.229169647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.229169647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−13.03143663237434, −12.90119877011867, −12.17645062688734, −11.55554332198250, −11.08301628111818, −10.80655593344666, −10.01839093743901, −9.517868476646001, −9.254013975375158, −8.971402287366922, −8.201148007526677, −7.795455286750519, −7.593599878550114, −6.916844176146069, −6.112571512594518, −5.731088099455178, −5.129627608179232, −4.749157530233660, −4.052281539232892, −3.580025035547990, −3.310912174928073, −2.383939351806920, −1.842297857673982, −1.060211048073994, −0.5708118324782255,
0.5708118324782255, 1.060211048073994, 1.842297857673982, 2.383939351806920, 3.310912174928073, 3.580025035547990, 4.052281539232892, 4.749157530233660, 5.129627608179232, 5.731088099455178, 6.112571512594518, 6.916844176146069, 7.593599878550114, 7.795455286750519, 8.201148007526677, 8.971402287366922, 9.254013975375158, 9.517868476646001, 10.01839093743901, 10.80655593344666, 11.08301628111818, 11.55554332198250, 12.17645062688734, 12.90119877011867, 13.03143663237434
