L(s) = 1 | + (0.923 + 0.382i)2-s + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (0.923 − 0.382i)5-s + (−0.923 + 0.382i)6-s + (0.382 + 0.923i)8-s − 1.00i·9-s + 10-s − 0.765·11-s − 12-s + (0.707 + 0.707i)13-s + (−0.382 + 0.923i)15-s + i·16-s + (0.382 − 0.923i)18-s + (0.923 + 0.382i)20-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (0.923 − 0.382i)5-s + (−0.923 + 0.382i)6-s + (0.382 + 0.923i)8-s − 1.00i·9-s + 10-s − 0.765·11-s − 12-s + (0.707 + 0.707i)13-s + (−0.382 + 0.923i)15-s + i·16-s + (0.382 − 0.923i)18-s + (0.923 + 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.771971157\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.771971157\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.923 + 0.382i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 0.765T + T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + 1.84iT - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 0.765iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 89 | \( 1 + 0.765iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.01311893890531292421213068726, −8.950260243622399241841978300250, −8.292284072281854212493352227975, −6.95113432790355447440706308222, −6.36533320991465972118048379281, −5.51663551428012382424857220621, −5.02932300600768411978431185198, −4.14203036302488975250800946887, −3.13590189295346853960459546997, −1.80732844300405348404991807916,
1.32573228339451575568794014122, 2.36763761140940274544485370535, 3.25363107092413021471558226961, 4.67804794510023469226798853657, 5.54620426870470469832882333026, 5.93770493961255733591015025030, 6.80379503062807381635410784481, 7.51438115367535812145682634824, 8.606258374700528102377797192023, 9.910050262615561939861646363434