| L(s) = 1 | − 0.0847·2-s − 2.25·3-s − 1.99·4-s − 0.792·5-s + 0.190·6-s + 0.338·8-s + 2.06·9-s + 0.0671·10-s + 4.82·11-s + 4.48·12-s − 3.03·13-s + 1.78·15-s + 3.95·16-s − 5.93·17-s − 0.175·18-s + 4.99·19-s + 1.57·20-s − 0.408·22-s + 2.59·23-s − 0.761·24-s − 4.37·25-s + 0.257·26-s + 2.10·27-s − 5.54·29-s − 0.151·30-s − 1.01·31-s − 1.01·32-s + ⋯ |
| L(s) = 1 | − 0.0599·2-s − 1.29·3-s − 0.996·4-s − 0.354·5-s + 0.0778·6-s + 0.119·8-s + 0.688·9-s + 0.0212·10-s + 1.45·11-s + 1.29·12-s − 0.842·13-s + 0.460·15-s + 0.989·16-s − 1.43·17-s − 0.0412·18-s + 1.14·19-s + 0.353·20-s − 0.0871·22-s + 0.540·23-s − 0.155·24-s − 0.874·25-s + 0.0505·26-s + 0.404·27-s − 1.03·29-s − 0.0275·30-s − 0.182·31-s − 0.178·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4830380884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4830380884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0847T + 2T^{2} \) |
| 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 + 0.792T + 5T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 - 2.59T + 23T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 - 2.40T + 37T^{2} \) |
| 41 | \( 1 + 7.84T + 41T^{2} \) |
| 43 | \( 1 - 4.98T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 - 5.31T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 83 | \( 1 + 2.63T + 83T^{2} \) |
| 89 | \( 1 - 9.86T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
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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
−8.772635739089542283364104757056, −7.50824865679589206091502024215, −7.06758904793963185807334314990, −6.06004435218731928715047155604, −5.55033389983533008002685997140, −4.58501335420715441104992765958, −4.26582708439579711578159650037, −3.20148113279640215422073720739, −1.59353533337509220969371877371, −0.45227277255109690125095110230,
0.45227277255109690125095110230, 1.59353533337509220969371877371, 3.20148113279640215422073720739, 4.26582708439579711578159650037, 4.58501335420715441104992765958, 5.55033389983533008002685997140, 6.06004435218731928715047155604, 7.06758904793963185807334314990, 7.50824865679589206091502024215, 8.772635739089542283364104757056
