L(s) = 1 | + 3-s − 2·7-s + 9-s − 4·11-s − 6·13-s − 6·17-s − 2·21-s − 4·23-s + 27-s + 4·29-s + 10·31-s − 4·33-s − 2·37-s − 6·39-s − 2·41-s − 8·43-s + 12·47-s − 3·49-s − 6·51-s + 12·53-s − 4·59-s + 2·61-s − 2·63-s − 4·67-s − 4·69-s − 4·71-s + 10·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.436·21-s − 0.834·23-s + 0.192·27-s + 0.742·29-s + 1.79·31-s − 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s − 0.840·51-s + 1.64·53-s − 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.488·67-s − 0.481·69-s − 0.474·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.131820915\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131820915\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.67836028948554475001548126371, −7.03797216527515100553413964284, −6.50157649314422002981040156045, −5.61696686066097488250891132837, −4.74878140873420572672293755763, −4.34325134938992504250556000135, −3.20859138627074280559035598943, −2.56433778811276467087411373879, −2.10849281981293727260394437440, −0.45397119587697427734426750720,
0.45397119587697427734426750720, 2.10849281981293727260394437440, 2.56433778811276467087411373879, 3.20859138627074280559035598943, 4.34325134938992504250556000135, 4.74878140873420572672293755763, 5.61696686066097488250891132837, 6.50157649314422002981040156045, 7.03797216527515100553413964284, 7.67836028948554475001548126371