L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.64 + 0.535i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.786 − 1.54i)6-s − 2.53·7-s + (0.707 − 0.707i)8-s + (2.42 + 1.76i)9-s + 1.00·10-s − 1.55·11-s + (−0.535 + 1.64i)12-s + (−1.11 − 1.11i)13-s + (1.79 + 1.79i)14-s + (−1.54 + 0.786i)15-s − 1.00·16-s + (−4.84 + 4.84i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.951 + 0.309i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.320 − 0.630i)6-s − 0.959·7-s + (0.250 − 0.250i)8-s + (0.808 + 0.587i)9-s + 0.316·10-s − 0.468·11-s + (−0.154 + 0.475i)12-s + (−0.309 − 0.309i)13-s + (0.479 + 0.479i)14-s + (−0.398 + 0.203i)15-s − 0.250·16-s + (−1.17 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3336785348\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3336785348\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.64 - 0.535i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (-3.85 - 4.70i)T \) |
good | 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 + (1.11 + 1.11i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.84 - 4.84i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.49 + 2.49i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.856 - 0.856i)T - 23iT^{2} \) |
| 29 | \( 1 + (5.19 + 5.19i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.70 + 1.70i)T - 31iT^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + (2.78 + 2.78i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.4iT - 47T^{2} \) |
| 53 | \( 1 + 5.16iT - 53T^{2} \) |
| 59 | \( 1 + (8.36 - 8.36i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.41 + 7.41i)T - 61iT^{2} \) |
| 67 | \( 1 - 1.98iT - 67T^{2} \) |
| 71 | \( 1 - 3.37iT - 71T^{2} \) |
| 73 | \( 1 + 0.480iT - 73T^{2} \) |
| 79 | \( 1 + (-6.52 - 6.52i)T + 79iT^{2} \) |
| 83 | \( 1 + 8.64iT - 83T^{2} \) |
| 89 | \( 1 + (10.5 + 10.5i)T + 89iT^{2} \) |
| 97 | \( 1 + (5.01 + 5.01i)T + 97iT^{2} \) |
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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
−10.05341952275112579565776485269, −9.492711861726393876953205045636, −8.564704583279415046806756725760, −8.010431011651040545913360099503, −7.06625655936887120146562149097, −6.21568808266862811779418908028, −4.63142840552106814281170032600, −3.76598793940258117098138835728, −2.90445995029745805650206508421, −2.03216175943198025036201372870,
0.14049134421063047807453264090, 1.92774332203192498284116359238, 3.04737345174291295424707916363, 4.15901200202830434584846815860, 5.23829055840093331055738303059, 6.58701506585689783567253104249, 7.00121497189273694718195475822, 7.925088496562756710851426616021, 8.687659625622054457522163580255, 9.324982984702778114119260879837