L(s) = 1 | + 3-s + 5-s − 2·9-s − 5·11-s − 13-s + 15-s − 3·17-s + 6·19-s − 6·23-s + 25-s − 5·27-s − 9·29-s − 5·33-s + 6·37-s − 39-s − 8·41-s + 6·43-s − 2·45-s − 3·47-s − 3·51-s − 12·53-s − 5·55-s + 6·57-s − 8·59-s + 4·61-s − 65-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s − 0.870·33-s + 0.986·37-s − 0.160·39-s − 1.24·41-s + 0.914·43-s − 0.298·45-s − 0.437·47-s − 0.420·51-s − 1.64·53-s − 0.674·55-s + 0.794·57-s − 1.04·59-s + 0.512·61-s − 0.124·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.860484777002540005828982336087, −7.74465662186675135601071019089, −7.69912855129361241271554591687, −6.28715004025144947864257791694, −5.55365867270428984617540263559, −4.85287109801086584074848656984, −3.56542759922310316766610018532, −2.71647306061986597321844269058, −1.93520907898928704565581532153, 0,
1.93520907898928704565581532153, 2.71647306061986597321844269058, 3.56542759922310316766610018532, 4.85287109801086584074848656984, 5.55365867270428984617540263559, 6.28715004025144947864257791694, 7.69912855129361241271554591687, 7.74465662186675135601071019089, 8.860484777002540005828982336087