L(s) = 1 | + (1 + i)2-s + (2.42 − 2.42i)3-s + 2i·4-s + (−1.75 − 1.39i)5-s + 4.84·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 8.74i·9-s + (−0.359 − 3.14i)10-s + (4.84 + 4.84i)12-s + (−2.42 + 2.42i)13-s + 3.74i·14-s + (−7.61 + 0.870i)15-s − 4·16-s + (8.74 − 8.74i)18-s + 4.21i·19-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (1.39 − 1.39i)3-s + i·4-s + (−0.782 − 0.622i)5-s + 1.97·6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 2.91i·9-s + (−0.113 − 0.993i)10-s + (1.39 + 1.39i)12-s + (−0.672 + 0.672i)13-s + 0.999i·14-s + (−1.96 + 0.224i)15-s − 16-s + (2.06 − 2.06i)18-s + 0.968i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.43361 - 0.143657i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.43361 - 0.143657i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (1.75 + 1.39i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 3 | \( 1 + (-2.42 + 2.42i)T - 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (2.42 - 2.42i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 4.21iT - 19T^{2} \) |
| 23 | \( 1 + (-0.741 + 0.741i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 0.625iT - 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 15.7iT - 79T^{2} \) |
| 83 | \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−12.14794406954378740469750074070, −11.70662474739020182850190586613, −9.219391919088838554355068134792, −8.580659276223495287429676220682, −7.84035246896200280733236761687, −7.24686236270078508543369096429, −6.03987452824794498748629078326, −4.54041860947147154718760251169, −3.29633260994323964361219565220, −1.95294742844658589481979869451,
2.50078484901365260922877867185, 3.43386252445853009353079482271, 4.34093375767852753195323479610, 5.08513957121305046705508251564, 7.23089473952283514547462027141, 8.155719404591099884764360641186, 9.276969049380823432237973426935, 10.26632109788117262864929386418, 10.76527228105043320103136378779, 11.56215376497432947204533684274