| L(s) = 1 | − 2-s − 0.111·3-s + 4-s − 2.69·5-s + 0.111·6-s − 8-s − 2.98·9-s + 2.69·10-s − 3.17·11-s − 0.111·12-s + 13-s + 0.300·15-s + 16-s − 0.810·17-s + 2.98·18-s − 4·19-s − 2.69·20-s + 3.17·22-s + 8.57·23-s + 0.111·24-s + 2.28·25-s − 26-s + 0.666·27-s + 3.17·29-s − 0.300·30-s − 3.57·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0642·3-s + 0.5·4-s − 1.20·5-s + 0.0454·6-s − 0.353·8-s − 0.995·9-s + 0.853·10-s − 0.957·11-s − 0.0321·12-s + 0.277·13-s + 0.0775·15-s + 0.250·16-s − 0.196·17-s + 0.704·18-s − 0.917·19-s − 0.603·20-s + 0.677·22-s + 1.78·23-s + 0.0227·24-s + 0.457·25-s − 0.196·26-s + 0.128·27-s + 0.589·29-s − 0.0548·30-s − 0.642·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5869443999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5869443999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.111T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 17 | \( 1 + 0.810T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8.57T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.39T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 0.378T + 67T^{2} \) |
| 71 | \( 1 + 2.36T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−9.552527937717444912730212481510, −8.632950970803048022971745079598, −8.201650019986468416907069995995, −7.43794042978314611688841527226, −6.57003698508000805537383170217, −5.53690031590369306204347945477, −4.52285955707805173394104176787, −3.34029596520883337536415853019, −2.46963818614324742099556100731, −0.60047185875958258134109976946,
0.60047185875958258134109976946, 2.46963818614324742099556100731, 3.34029596520883337536415853019, 4.52285955707805173394104176787, 5.53690031590369306204347945477, 6.57003698508000805537383170217, 7.43794042978314611688841527226, 8.201650019986468416907069995995, 8.632950970803048022971745079598, 9.552527937717444912730212481510
