| L(s) = 1 | + 3-s + 4·7-s − 2·9-s + 3·11-s − 2·13-s − 17-s − 19-s + 4·21-s + 23-s − 5·27-s + 8·31-s + 3·33-s − 2·37-s − 2·39-s + 41-s + 12·43-s + 6·47-s + 9·49-s − 51-s − 4·53-s − 57-s + 12·59-s − 8·63-s + 13·67-s + 69-s + 12·71-s − 17·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.904·11-s − 0.554·13-s − 0.242·17-s − 0.229·19-s + 0.872·21-s + 0.208·23-s − 0.962·27-s + 1.43·31-s + 0.522·33-s − 0.328·37-s − 0.320·39-s + 0.156·41-s + 1.82·43-s + 0.875·47-s + 9/7·49-s − 0.140·51-s − 0.549·53-s − 0.132·57-s + 1.56·59-s − 1.00·63-s + 1.58·67-s + 0.120·69-s + 1.42·71-s − 1.98·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.879311261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.879311261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.390902451562958491957440349762, −7.73570175095482955640696517468, −7.03454162395028209903297604348, −6.09771145409907104404464047508, −5.30866601482743800419366313081, −4.52378274870349201159710526000, −3.87866168934077473105692873502, −2.71690637658407031705062332425, −2.05547272414673488375387186861, −0.954736524589207365541990687901,
0.954736524589207365541990687901, 2.05547272414673488375387186861, 2.71690637658407031705062332425, 3.87866168934077473105692873502, 4.52378274870349201159710526000, 5.30866601482743800419366313081, 6.09771145409907104404464047508, 7.03454162395028209903297604348, 7.73570175095482955640696517468, 8.390902451562958491957440349762
