| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 5·11-s − 4·13-s + 14-s + 16-s + 6·19-s − 20-s + 5·22-s − 4·25-s + 4·26-s − 28-s − 3·29-s + 5·31-s − 32-s + 35-s − 8·37-s − 6·38-s + 40-s + 6·41-s − 5·44-s − 2·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.50·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s + 1.06·22-s − 4/5·25-s + 0.784·26-s − 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s + 0.169·35-s − 1.31·37-s − 0.973·38-s + 0.158·40-s + 0.937·41-s − 0.753·44-s − 0.291·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−15.38921213919544, −14.80799477701153, −14.04900457254560, −13.65858192527938, −12.95426120809857, −12.45169654500182, −11.98173321722349, −11.44538368218230, −10.88832743393654, −10.22571651794019, −9.875636274769044, −9.437719434774115, −8.717343030879652, −8.027599500456282, −7.629228548216291, −7.296943123197613, −6.606393290184113, −5.773579412730795, −5.269940969781301, −4.733062147138039, −3.786680563110014, −3.098819533103983, −2.565380845133735, −1.859358638622066, −0.7176106762563894, 0,
0.7176106762563894, 1.859358638622066, 2.565380845133735, 3.098819533103983, 3.786680563110014, 4.733062147138039, 5.269940969781301, 5.773579412730795, 6.606393290184113, 7.296943123197613, 7.629228548216291, 8.027599500456282, 8.717343030879652, 9.437719434774115, 9.875636274769044, 10.22571651794019, 10.88832743393654, 11.44538368218230, 11.98173321722349, 12.45169654500182, 12.95426120809857, 13.65858192527938, 14.04900457254560, 14.80799477701153, 15.38921213919544
