L(s) = 1 | + 2·3-s − 7-s + 9-s − 6·11-s + 4·13-s − 2·17-s − 4·19-s − 2·21-s + 23-s − 4·27-s − 6·29-s + 4·31-s − 12·33-s + 4·37-s + 8·39-s − 2·41-s + 6·43-s + 49-s − 4·51-s + 8·53-s − 8·57-s + 6·59-s − 6·61-s − 63-s − 10·67-s + 2·69-s + 14·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.436·21-s + 0.208·23-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 2.08·33-s + 0.657·37-s + 1.28·39-s − 0.312·41-s + 0.914·43-s + 1/7·49-s − 0.560·51-s + 1.09·53-s − 1.05·57-s + 0.781·59-s − 0.768·61-s − 0.125·63-s − 1.22·67-s + 0.240·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.23231806128060, −12.79554169909140, −12.35415383357575, −11.53577745698266, −11.14132039427693, −10.70155020428402, −10.27779585873793, −9.803363889218551, −9.182997843212868, −8.832772861908701, −8.371920806814750, −8.046029096062200, −7.533761194143743, −7.091840240284566, −6.421568137051262, −5.803797864076614, −5.598979873962182, −4.763206032638863, −4.248060190351016, −3.718746006901063, −3.171692074317227, −2.658933133489954, −2.285974172820584, −1.708325998870330, −0.7352375707585642, 0,
0.7352375707585642, 1.708325998870330, 2.285974172820584, 2.658933133489954, 3.171692074317227, 3.718746006901063, 4.248060190351016, 4.763206032638863, 5.598979873962182, 5.803797864076614, 6.421568137051262, 7.091840240284566, 7.533761194143743, 8.046029096062200, 8.371920806814750, 8.832772861908701, 9.182997843212868, 9.803363889218551, 10.27779585873793, 10.70155020428402, 11.14132039427693, 11.53577745698266, 12.35415383357575, 12.79554169909140, 13.23231806128060