| L(s) = 1 | + 3-s + 2.37·7-s + 9-s + 2.37·11-s + 13-s − 4.37·17-s + 4.74·19-s + 2.37·21-s + 6.37·23-s + 27-s − 2·29-s − 4.74·31-s + 2.37·33-s + 0.372·37-s + 39-s + 4.37·41-s + 4·43-s + 12.7·47-s − 1.37·49-s − 4.37·51-s + 3.62·53-s + 4.74·57-s − 8·59-s − 9.11·61-s + 2.37·63-s + 4·67-s + 6.37·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.896·7-s + 0.333·9-s + 0.715·11-s + 0.277·13-s − 1.06·17-s + 1.08·19-s + 0.517·21-s + 1.32·23-s + 0.192·27-s − 0.371·29-s − 0.852·31-s + 0.412·33-s + 0.0612·37-s + 0.160·39-s + 0.682·41-s + 0.609·43-s + 1.85·47-s − 0.196·49-s − 0.612·51-s + 0.498·53-s + 0.628·57-s − 1.04·59-s − 1.16·61-s + 0.298·63-s + 0.488·67-s + 0.767·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.335783724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.335783724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 0.372T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.62T + 71T^{2} \) |
| 73 | \( 1 + 7.48T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
| 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
−7.69572137720768365029174925304, −7.38679197514102369397885204861, −6.57945066594830394329162263666, −5.73566867663928309763927637232, −4.94157706104607104078745913028, −4.28474591326562358354814137913, −3.54279915325554017676907231381, −2.66057393436804968087951685845, −1.76916228680782987145304880796, −0.945597211628375946423849223533,
0.945597211628375946423849223533, 1.76916228680782987145304880796, 2.66057393436804968087951685845, 3.54279915325554017676907231381, 4.28474591326562358354814137913, 4.94157706104607104078745913028, 5.73566867663928309763927637232, 6.57945066594830394329162263666, 7.38679197514102369397885204861, 7.69572137720768365029174925304
