L(s) = 1 | − 1.41·5-s + 9-s + 1.41·13-s + 1.41·17-s + 1.00·25-s − 1.41·41-s − 1.41·45-s − 2·53-s + 1.41·61-s − 2.00·65-s − 1.41·73-s + 81-s − 2.00·85-s + 1.41·89-s − 1.41·97-s + 1.41·101-s − 2·113-s + 1.41·117-s + ⋯ |
L(s) = 1 | − 1.41·5-s + 9-s + 1.41·13-s + 1.41·17-s + 1.00·25-s − 1.41·41-s − 1.41·45-s − 2·53-s + 1.41·61-s − 2.00·65-s − 1.41·73-s + 81-s − 2.00·85-s + 1.41·89-s − 1.41·97-s + 1.41·101-s − 2·113-s + 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8823605532\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8823605532\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 2T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 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
−10.56340104363766966345395249230, −9.737142218688120064088862898415, −8.587255073573434981989680554681, −7.928162518312773339060123037204, −7.21483070278693134140077451141, −6.22911048917339716285301823291, −4.97640589235543056838139389236, −3.92279196047754250672830626475, −3.34909874627217355643005251053, −1.32096372088368867834205028680,
1.32096372088368867834205028680, 3.34909874627217355643005251053, 3.92279196047754250672830626475, 4.97640589235543056838139389236, 6.22911048917339716285301823291, 7.21483070278693134140077451141, 7.928162518312773339060123037204, 8.587255073573434981989680554681, 9.737142218688120064088862898415, 10.56340104363766966345395249230