L(s) = 1 | + 2-s − 3-s + 4-s − 4.15·5-s − 6-s + 8-s + 9-s − 4.15·10-s + 5.67·11-s − 12-s + 6.21·13-s + 4.15·15-s + 16-s − 5.61·17-s + 18-s − 19-s − 4.15·20-s + 5.67·22-s − 1.65·23-s − 24-s + 12.2·25-s + 6.21·26-s − 27-s + 2.70·29-s + 4.15·30-s − 3.74·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.85·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.31·10-s + 1.71·11-s − 0.288·12-s + 1.72·13-s + 1.07·15-s + 0.250·16-s − 1.36·17-s + 0.235·18-s − 0.229·19-s − 0.929·20-s + 1.21·22-s − 0.345·23-s − 0.204·24-s + 2.45·25-s + 1.21·26-s − 0.192·27-s + 0.503·29-s + 0.759·30-s − 0.673·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.892915608\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892915608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 4.15T + 5T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 13 | \( 1 - 6.21T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 8.86T + 61T^{2} \) |
| 67 | \( 1 - 3.39T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 + 0.610T + 97T^{2} \) |
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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
−8.164018466449385168078447617149, −7.15715828818247646356789056207, −6.61795650210540606130163868254, −6.21681212638821527368248763547, −5.05537175968073336084220117792, −4.23466252800828587847633086592, −3.87687978260553707519381937581, −3.34371653406661416881212603134, −1.76575581703738971465047641933, −0.70272824203797912375097795882,
0.70272824203797912375097795882, 1.76575581703738971465047641933, 3.34371653406661416881212603134, 3.87687978260553707519381937581, 4.23466252800828587847633086592, 5.05537175968073336084220117792, 6.21681212638821527368248763547, 6.61795650210540606130163868254, 7.15715828818247646356789056207, 8.164018466449385168078447617149