| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 4·11-s − 2·13-s + 16-s + 8·17-s + 6·19-s + 20-s − 4·22-s + 4·23-s + 25-s + 2·26-s + 6·29-s − 4·31-s − 32-s − 8·34-s − 10·37-s − 6·38-s − 40-s + 4·41-s + 4·43-s + 4·44-s − 4·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.94·17-s + 1.37·19-s + 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.37·34-s − 1.64·37-s − 0.973·38-s − 0.158·40-s + 0.624·41-s + 0.609·43-s + 0.603·44-s − 0.589·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876808820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876808820\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.453534801851306984124575307061, −7.48291094929870202480489977046, −7.18825155034560790075804539710, −6.21000403989565597872440160728, −5.54829591049291655462745991509, −4.76471141305607830172592166766, −3.49757290094191532828667451027, −2.94048924601458656354928461744, −1.61462776166642999783063348501, −0.946860701606855831609790951771,
0.946860701606855831609790951771, 1.61462776166642999783063348501, 2.94048924601458656354928461744, 3.49757290094191532828667451027, 4.76471141305607830172592166766, 5.54829591049291655462745991509, 6.21000403989565597872440160728, 7.18825155034560790075804539710, 7.48291094929870202480489977046, 8.453534801851306984124575307061
