L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s + 2·13-s − 15-s − 7·17-s + 2·19-s − 21-s + 25-s + 27-s + 9·29-s − 7·31-s + 33-s + 35-s − 37-s + 2·39-s − 11·41-s + 11·43-s − 45-s − 8·47-s − 6·49-s − 7·51-s − 11·53-s − 55-s + 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 1.69·17-s + 0.458·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.67·29-s − 1.25·31-s + 0.174·33-s + 0.169·35-s − 0.164·37-s + 0.320·39-s − 1.71·41-s + 1.67·43-s − 0.149·45-s − 1.16·47-s − 6/7·49-s − 0.980·51-s − 1.51·53-s − 0.134·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.36284080651633760814115965096, −6.71531747813583648577899590870, −6.31386701024794796752254330660, −5.19476649920357895611225797309, −4.49848124902179781710662729827, −3.77312617670668562837690614547, −3.14349247078851789679846180074, −2.27728968477591729264447575446, −1.30793321440505781480997690875, 0,
1.30793321440505781480997690875, 2.27728968477591729264447575446, 3.14349247078851789679846180074, 3.77312617670668562837690614547, 4.49848124902179781710662729827, 5.19476649920357895611225797309, 6.31386701024794796752254330660, 6.71531747813583648577899590870, 7.36284080651633760814115965096