| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 11-s + 2·14-s + 16-s + 6·17-s + 4·19-s + 22-s − 6·23-s − 5·25-s − 2·28-s − 6·29-s − 8·31-s − 32-s − 6·34-s + 10·37-s − 4·38-s + 6·41-s + 8·43-s − 44-s + 6·46-s − 6·47-s − 3·49-s + 5·50-s + 2·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.301·11-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.213·22-s − 1.25·23-s − 25-s − 0.377·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001454397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001454397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.04363582000027, −14.44754818358316, −14.12484746063238, −13.31514951113640, −12.82264002874983, −12.36172567474429, −11.78532070889138, −11.17955235529549, −10.78985925881790, −9.799909255033646, −9.703612732069229, −9.432571788426515, −8.442570347209899, −7.785493960601775, −7.619869647862185, −6.946883220601598, −6.021049346661547, −5.798455937185703, −5.193982644642716, −4.008613540331017, −3.649741420598570, −2.873144174058457, −2.173884432040065, −1.340373420147414, −0.4373436428150299,
0.4373436428150299, 1.340373420147414, 2.173884432040065, 2.873144174058457, 3.649741420598570, 4.008613540331017, 5.193982644642716, 5.798455937185703, 6.021049346661547, 6.946883220601598, 7.619869647862185, 7.785493960601775, 8.442570347209899, 9.432571788426515, 9.703612732069229, 9.799909255033646, 10.78985925881790, 11.17955235529549, 11.78532070889138, 12.36172567474429, 12.82264002874983, 13.31514951113640, 14.12484746063238, 14.44754818358316, 15.04363582000027
