L(s) = 1 | + 3-s − 3·5-s + 9-s − 3·15-s − 17-s + 4·23-s + 4·25-s + 27-s + 3·29-s − 8·31-s + 5·37-s − 3·41-s + 4·43-s − 3·45-s + 8·47-s − 7·49-s − 51-s − 13·53-s − 12·59-s + 15·61-s − 12·67-s + 4·69-s − 8·71-s − 3·73-s + 4·75-s − 4·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.774·15-s − 0.242·17-s + 0.834·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s − 1.43·31-s + 0.821·37-s − 0.468·41-s + 0.609·43-s − 0.447·45-s + 1.16·47-s − 49-s − 0.140·51-s − 1.78·53-s − 1.56·59-s + 1.92·61-s − 1.46·67-s + 0.481·69-s − 0.949·71-s − 0.351·73-s + 0.461·75-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.000566271266429576510774711830, −7.47234443540029826033922661903, −6.87649985850630835513411278567, −5.88202043925037385618298963778, −4.80514252478158306801851666730, −4.19411966750433846265189025747, −3.41224769444100026837217865227, −2.69482322049120329533779377296, −1.39097143481885923696838971947, 0,
1.39097143481885923696838971947, 2.69482322049120329533779377296, 3.41224769444100026837217865227, 4.19411966750433846265189025747, 4.80514252478158306801851666730, 5.88202043925037385618298963778, 6.87649985850630835513411278567, 7.47234443540029826033922661903, 8.000566271266429576510774711830