L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)8-s − i·9-s + 1.00i·10-s − 1.41·11-s + (0.707 − 0.707i)13-s − 1.00·16-s + (−1 − i)17-s + (0.707 − 0.707i)18-s + 1.41i·19-s + (−0.707 + 0.707i)20-s + (−1.00 − 1.00i)22-s + (1 − i)23-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)8-s − i·9-s + 1.00i·10-s − 1.41·11-s + (0.707 − 0.707i)13-s − 1.00·16-s + (−1 − i)17-s + (0.707 − 0.707i)18-s + 1.41i·19-s + (−0.707 + 0.707i)20-s + (−1.00 − 1.00i)22-s + (1 − i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.284979530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284979530\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−11.23860949860926230438314408946, −10.51476870168407939106194563146, −9.421869707721754023159928666661, −8.437814325319654506358770891546, −7.46663438111658628570482220175, −6.52796591082720012090192753083, −5.85165018981928677886883756514, −4.88075210218644157246490173372, −3.44362123516619136297043713273, −2.59725577628366321846094181049,
1.74003422160462123275893510149, 2.73577770830392929720643916186, 4.38009290165157163359683087229, 5.09938706106131747473121695398, 5.90463045294117150999879511780, 7.10417125949962314076499153100, 8.528397634556543322923615490025, 9.195922682456449502399551358261, 10.35619932850162933897762526673, 10.86235401609738390880396142256