| L(s) = 1 | + 0.402·3-s − 5-s + 7-s − 2.83·9-s + 1.78·11-s − 13-s − 0.402·15-s + 7.18·17-s − 5.62·19-s + 0.402·21-s + 1.08·23-s + 25-s − 2.34·27-s − 8.01·29-s − 7.90·31-s + 0.718·33-s − 35-s − 9.53·37-s − 0.402·39-s + 1.73·41-s + 10.9·43-s + 2.83·45-s + 0.542·47-s + 49-s + 2.89·51-s + 6.30·53-s − 1.78·55-s + ⋯ |
| L(s) = 1 | + 0.232·3-s − 0.447·5-s + 0.377·7-s − 0.945·9-s + 0.538·11-s − 0.277·13-s − 0.103·15-s + 1.74·17-s − 1.29·19-s + 0.0878·21-s + 0.226·23-s + 0.200·25-s − 0.452·27-s − 1.48·29-s − 1.41·31-s + 0.125·33-s − 0.169·35-s − 1.56·37-s − 0.0644·39-s + 0.270·41-s + 1.67·43-s + 0.423·45-s + 0.0791·47-s + 0.142·49-s + 0.404·51-s + 0.866·53-s − 0.240·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.402T + 3T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 - 1.08T + 23T^{2} \) |
| 29 | \( 1 + 8.01T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 9.53T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.542T + 47T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 2.71T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 0.819T + 79T^{2} \) |
| 83 | \( 1 + 0.910T + 83T^{2} \) |
| 89 | \( 1 + 9.09T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.088959564185993404114365656417, −7.59057847533612610582993564964, −6.81258342349936927036531829118, −5.70053216603378096346573738263, −5.36274315641329399006502804941, −4.10292478336619107641313114002, −3.56280013296045319663965793437, −2.55634005837277883175771090461, −1.48786643616585563309624968970, 0,
1.48786643616585563309624968970, 2.55634005837277883175771090461, 3.56280013296045319663965793437, 4.10292478336619107641313114002, 5.36274315641329399006502804941, 5.70053216603378096346573738263, 6.81258342349936927036531829118, 7.59057847533612610582993564964, 8.088959564185993404114365656417
