L(s) = 1 | + 1.41·3-s + i·5-s + (−0.707 + 0.707i)7-s + 1.00·9-s + 1.41i·15-s + (−1.00 + 1.00i)21-s − 1.41i·23-s − 25-s + 2·29-s + (−0.707 − 0.707i)35-s + 2i·41-s − 1.41i·43-s + 1.00i·45-s − 1.41·47-s − 1.00i·49-s + ⋯ |
L(s) = 1 | + 1.41·3-s + i·5-s + (−0.707 + 0.707i)7-s + 1.00·9-s + 1.41i·15-s + (−1.00 + 1.00i)21-s − 1.41i·23-s − 25-s + 2·29-s + (−0.707 − 0.707i)35-s + 2i·41-s − 1.41i·43-s + 1.00i·45-s − 1.41·47-s − 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.504609325\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.504609325\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 2iT - T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.951072030128261790860211124697, −9.322581398509889351506150269757, −8.414677168539626117200315587999, −7.930783705846155493843864390680, −6.69537744581888639462228010301, −6.31935010476225589123642825744, −4.81691452895326005921719859864, −3.54039684525809606420512015719, −2.89709334595176181591152831215, −2.18861563401523290385669901959,
1.35849802328338712209255188712, 2.74726002137464069677077475211, 3.66937449300907883328296566916, 4.45278513780856770361299141264, 5.62460238959048229618598604484, 6.81632344585498536511998192800, 7.68896527229820189225773134837, 8.351414923921519626933471680704, 9.094539555409779792965332100456, 9.693164450475603469305425520211