| L(s) = 1 | − 2-s + 3-s + 4-s − 1.23·5-s − 6-s − 8-s + 9-s + 1.23·10-s − 2·11-s + 12-s − 5.23·13-s − 1.23·15-s + 16-s − 2.47·17-s − 18-s + 19-s − 1.23·20-s + 2·22-s − 5.70·23-s − 24-s − 3.47·25-s + 5.23·26-s + 27-s + 8.47·29-s + 1.23·30-s − 7.70·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.552·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.390·10-s − 0.603·11-s + 0.288·12-s − 1.45·13-s − 0.319·15-s + 0.250·16-s − 0.599·17-s − 0.235·18-s + 0.229·19-s − 0.276·20-s + 0.426·22-s − 1.19·23-s − 0.204·24-s − 0.694·25-s + 1.02·26-s + 0.192·27-s + 1.57·29-s + 0.225·30-s − 1.38·31-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
| \(L(1)\) |
\(\approx\) |
\(0.9567182800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9567182800\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.139099733009364042678932458274, −7.53141473339857203965850023120, −7.10657446445935161922039510827, −6.15704575510189514846040835793, −5.22233504922285046673103937275, −4.39293060188789819963826580096, −3.58094665258234942390189563025, −2.53875079971129584637251026987, −2.06676909767428551216194700148, −0.53321215643479004058923672555,
0.53321215643479004058923672555, 2.06676909767428551216194700148, 2.53875079971129584637251026987, 3.58094665258234942390189563025, 4.39293060188789819963826580096, 5.22233504922285046673103937275, 6.15704575510189514846040835793, 7.10657446445935161922039510827, 7.53141473339857203965850023120, 8.139099733009364042678932458274
