L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·5-s + 1.41·6-s + 7-s − 8-s + 1.00·9-s − 1.41·10-s − 1.41·12-s − 14-s − 2.00·15-s + 16-s − 1.41·17-s − 1.00·18-s + 1.41·20-s − 1.41·21-s + 23-s + 1.41·24-s + 1.00·25-s + 28-s + 2.00·30-s + 1.41·31-s − 32-s + 1.41·34-s + 1.41·35-s + 1.00·36-s + ⋯ |
L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·5-s + 1.41·6-s + 7-s − 8-s + 1.00·9-s − 1.41·10-s − 1.41·12-s − 14-s − 2.00·15-s + 16-s − 1.41·17-s − 1.00·18-s + 1.41·20-s − 1.41·21-s + 23-s + 1.41·24-s + 1.00·25-s + 28-s + 2.00·30-s + 1.41·31-s − 32-s + 1.41·34-s + 1.41·35-s + 1.00·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5415160633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5415160633\) |
\(L(1)\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−10.71746654881213849393467145918, −10.11464768694692600273623315811, −9.145650947679682051419535950898, −8.343720507318296631525673333154, −6.98491011999310771897646392505, −6.38582717846128897832791342044, −5.53716188043293091327640433004, −4.73683732226302747363667708072, −2.44541519955167362622504806431, −1.31028116421492288237319523058,
1.31028116421492288237319523058, 2.44541519955167362622504806431, 4.73683732226302747363667708072, 5.53716188043293091327640433004, 6.38582717846128897832791342044, 6.98491011999310771897646392505, 8.343720507318296631525673333154, 9.145650947679682051419535950898, 10.11464768694692600273623315811, 10.71746654881213849393467145918