L(s) = 1 | + 1.41·3-s − 5-s + 1.41·7-s + 1.00·9-s − 1.41·15-s + 2.00·21-s − 1.41·23-s + 25-s − 1.41·35-s − 1.41·43-s − 1.00·45-s + 1.41·47-s + 1.00·49-s + 2·61-s + 1.41·63-s − 1.41·67-s − 2.00·69-s + 1.41·75-s − 0.999·81-s − 1.41·83-s − 2·89-s − 1.41·103-s − 2.00·105-s + 1.41·107-s − 2·109-s + 1.41·115-s + ⋯ |
L(s) = 1 | + 1.41·3-s − 5-s + 1.41·7-s + 1.00·9-s − 1.41·15-s + 2.00·21-s − 1.41·23-s + 25-s − 1.41·35-s − 1.41·43-s − 1.00·45-s + 1.41·47-s + 1.00·49-s + 2·61-s + 1.41·63-s − 1.41·67-s − 2.00·69-s + 1.41·75-s − 0.999·81-s − 1.41·83-s − 2·89-s − 1.41·103-s − 2.00·105-s + 1.41·107-s − 2·109-s + 1.41·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.602216213\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.602216213\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.726596552189099725140848056567, −8.664963403798591759347843478425, −8.295998040127648597193030973554, −7.73697928437011101205136340001, −6.99155893631541424883519734617, −5.51635208355382842977662266503, −4.40751705592858610362505647469, −3.82926753282081635255589313073, −2.71468546190314114139779289000, −1.65679114841356520587875148021,
1.65679114841356520587875148021, 2.71468546190314114139779289000, 3.82926753282081635255589313073, 4.40751705592858610362505647469, 5.51635208355382842977662266503, 6.99155893631541424883519734617, 7.73697928437011101205136340001, 8.295998040127648597193030973554, 8.664963403798591759347843478425, 9.726596552189099725140848056567