L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·6-s − 8-s + 1.00·9-s − 1.41·12-s + 16-s + 1.41·17-s − 1.00·18-s + 1.41·19-s + 1.41·24-s + 25-s − 32-s − 1.41·34-s + 1.00·36-s − 1.41·38-s − 1.41·41-s − 1.41·48-s − 50-s − 2.00·51-s − 2.00·57-s + 1.41·59-s + 64-s − 2·67-s + 1.41·68-s − 1.00·72-s + ⋯ |
L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·6-s − 8-s + 1.00·9-s − 1.41·12-s + 16-s + 1.41·17-s − 1.00·18-s + 1.41·19-s + 1.41·24-s + 25-s − 32-s − 1.41·34-s + 1.00·36-s − 1.41·38-s − 1.41·41-s − 1.41·48-s − 50-s − 2.00·51-s − 2.00·57-s + 1.41·59-s + 64-s − 2·67-s + 1.41·68-s − 1.00·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3708581998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3708581998\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 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
−11.60850772798744020230300107710, −10.47532312828152730033100519341, −10.00930037858288919563303014591, −8.881218437890591878014609033370, −7.69499955236362975709200795848, −6.88355989854771910773220585407, −5.86342121267483289858218510497, −5.10615939355887923329439717519, −3.19680226230930867537077590072, −1.16546286981075352074177568114,
1.16546286981075352074177568114, 3.19680226230930867537077590072, 5.10615939355887923329439717519, 5.86342121267483289858218510497, 6.88355989854771910773220585407, 7.69499955236362975709200795848, 8.881218437890591878014609033370, 10.00930037858288919563303014591, 10.47532312828152730033100519341, 11.60850772798744020230300107710