L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.382 − 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−1.30 + 1.30i)13-s + i·15-s + 0.765i·19-s + (0.382 + 0.923i)21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.382 + 0.923i)35-s + (1.70 − 0.707i)39-s + (0.382 − 0.923i)45-s + 1.00i·49-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.382 − 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−1.30 + 1.30i)13-s + i·15-s + 0.765i·19-s + (0.382 + 0.923i)21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.382 + 0.923i)35-s + (1.70 − 0.707i)39-s + (0.382 − 0.923i)45-s + 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5217809761\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5217809761\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 + 0.382i)T \) |
| 5 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - 0.765iT - T^{2} \) |
| 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 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 - 1.84T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 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
−8.951788769498723536009461388451, −7.906833404315944964374404582791, −7.22270600699145042286391603407, −6.80597010404950438154875051574, −5.80487158921769292527802981935, −5.02650582056583415426351446829, −4.39339357819245294997815385609, −3.58508307847642596690946181672, −2.07483691843020813095914769882, −1.00991624650945446467931092421,
0.42614791313030668874294794406, 2.52621335711375944197659963931, 3.05530646386958373743540368208, 4.11205462903971628560263862571, 5.11441074311235905104273159083, 5.60782853870331109566493749940, 6.65812395965823788660405176834, 6.93312311874782241439005281964, 7.86100885711562542914360694185, 8.838073353409543104981287376248