L(s) = 1 | − i·2-s − 4-s + (−0.707 + 0.707i)5-s + i·8-s − 9-s + (0.707 + 0.707i)10-s + 1.41i·13-s + 16-s + 1.41i·17-s + i·18-s + (0.707 − 0.707i)20-s − 1.00i·25-s + 1.41·26-s − i·32-s + 1.41·34-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−0.707 + 0.707i)5-s + i·8-s − 9-s + (0.707 + 0.707i)10-s + 1.41i·13-s + 16-s + 1.41i·17-s + i·18-s + (0.707 − 0.707i)20-s − 1.00i·25-s + 1.41·26-s − i·32-s + 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5568104354\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5568104354\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 2iT - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−10.46238853933234341838056300457, −9.667332529496313533860220420156, −8.544680885048340428591286843487, −8.240522059169689902813379734440, −6.89604458435717238916698407724, −6.01430082940369843112638691450, −4.74815458656768011716886465042, −3.83947385823145028259015544178, −3.02755718847270443124899092917, −1.84120496562006941454486017563,
0.53534013729196280025652651376, 2.98530253044838178819832232157, 4.01903251288157073759077382382, 5.30304970252868633953811365137, 5.45404758597821470960295222444, 6.84961743495114197441092533626, 7.68497850518753325857033312683, 8.321375610689721246121577038221, 8.983558221277207994782848661729, 9.807968636322277987689931640934