L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 3·11-s + 12-s − 4·13-s + 16-s + 18-s + 4·19-s + 3·22-s + 24-s − 4·26-s + 27-s + 9·29-s + 31-s + 32-s + 3·33-s + 36-s − 8·37-s + 4·38-s − 4·39-s + 10·43-s + 3·44-s − 6·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.639·22-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 1.67·29-s + 0.179·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 1.31·37-s + 0.648·38-s − 0.640·39-s + 1.52·43-s + 0.452·44-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.489827714\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.489827714\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.78527528560740467165450102421, −7.08796827540644102346429459136, −6.64571139957631123674471614273, −5.72346223923502586275603125591, −4.94440216389551714616695101290, −4.35097518834865486146265864833, −3.51686649609977409262617892019, −2.83968090791970381564132606222, −2.02920537976508949653146175551, −0.964617224110125412344319753656,
0.964617224110125412344319753656, 2.02920537976508949653146175551, 2.83968090791970381564132606222, 3.51686649609977409262617892019, 4.35097518834865486146265864833, 4.94440216389551714616695101290, 5.72346223923502586275603125591, 6.64571139957631123674471614273, 7.08796827540644102346429459136, 7.78527528560740467165450102421