| L(s) = 1 | + 16·2-s − 34.9·3-s + 256·4-s − 530.·5-s − 559.·6-s − 1.22e4·7-s + 4.09e3·8-s − 1.84e4·9-s − 8.48e3·10-s − 5.23e4·11-s − 8.95e3·12-s − 1.95e5·14-s + 1.85e4·15-s + 6.55e4·16-s + 5.23e5·17-s − 2.95e5·18-s − 8.13e5·19-s − 1.35e5·20-s + 4.28e5·21-s − 8.37e5·22-s − 7.50e5·23-s − 1.43e5·24-s − 1.67e6·25-s + 1.33e6·27-s − 3.13e6·28-s − 5.53e6·29-s + 2.96e5·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.249·3-s + 0.5·4-s − 0.379·5-s − 0.176·6-s − 1.92·7-s + 0.353·8-s − 0.937·9-s − 0.268·10-s − 1.07·11-s − 0.124·12-s − 1.36·14-s + 0.0946·15-s + 0.250·16-s + 1.52·17-s − 0.663·18-s − 1.43·19-s − 0.189·20-s + 0.480·21-s − 0.761·22-s − 0.558·23-s − 0.0881·24-s − 0.856·25-s + 0.483·27-s − 0.963·28-s − 1.45·29-s + 0.0669·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1344154368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1344154368\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 34.9T + 1.96e4T^{2} \) |
| 5 | \( 1 + 530.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.22e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.23e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 5.23e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.13e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.50e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.53e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.25e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.72e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.16e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.28e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.36e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.54e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.30e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.47e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.48e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 9.00e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.35e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.24e8T + 7.60e17T^{2} \) |
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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
−10.20906747566839891598445208423, −9.162047261119402300505411435295, −7.941890488781273245435628113449, −6.98283060725489290629793645671, −5.88754493414919385383296124241, −5.47522307407225278786007706604, −3.78890171142248074649895493944, −3.25847839685020593392624469673, −2.16302754223686635340987651118, −0.13054512813957992944594851539,
0.13054512813957992944594851539, 2.16302754223686635340987651118, 3.25847839685020593392624469673, 3.78890171142248074649895493944, 5.47522307407225278786007706604, 5.88754493414919385383296124241, 6.98283060725489290629793645671, 7.941890488781273245435628113449, 9.162047261119402300505411435295, 10.20906747566839891598445208423
