L(s) = 1 | + (1.83 − 1.27i)5-s − 0.920·7-s + 4.88·11-s − 0.822i·13-s − 5.93i·17-s − 7.43i·19-s + (2.12 + 4.29i)23-s + (1.75 − 4.68i)25-s + 2.63i·29-s + 5.24·31-s + (−1.69 + 1.17i)35-s − 7.23·37-s − 1.93i·41-s + 1.42·43-s − 13.2·47-s + ⋯ |
L(s) = 1 | + (0.821 − 0.569i)5-s − 0.347·7-s + 1.47·11-s − 0.228i·13-s − 1.43i·17-s − 1.70i·19-s + (0.443 + 0.896i)23-s + (0.350 − 0.936i)25-s + 0.488i·29-s + 0.941·31-s + (−0.285 + 0.198i)35-s − 1.18·37-s − 0.301i·41-s + 0.217·43-s − 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.227929513\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.227929513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.83 + 1.27i)T \) |
| 23 | \( 1 + (-2.12 - 4.29i)T \) |
good | 7 | \( 1 + 0.920T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 0.822iT - 13T^{2} \) |
| 17 | \( 1 + 5.93iT - 17T^{2} \) |
| 19 | \( 1 + 7.43iT - 19T^{2} \) |
| 29 | \( 1 - 2.63iT - 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 + 1.93iT - 41T^{2} \) |
| 43 | \( 1 - 1.42T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 0.0569iT - 53T^{2} \) |
| 59 | \( 1 + 2.50iT - 59T^{2} \) |
| 61 | \( 1 - 5.95iT - 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 + 8.61iT - 71T^{2} \) |
| 73 | \( 1 - 3.83iT - 73T^{2} \) |
| 79 | \( 1 - 5.42iT - 79T^{2} \) |
| 83 | \( 1 + 1.52iT - 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.528704115925660888811458962046, −7.30564250069470430599335697643, −6.75464605942904174771816777538, −6.15356022866449451091036729266, −5.04207174166276388672326087824, −4.79345489475214540829469418806, −3.52664668355176439151517433165, −2.74528810942945675140741171774, −1.60761459007167959116718234904, −0.65206479458566418174700533289,
1.36391992190738093686017882265, 2.02503686888088496975828805662, 3.29850739635160258775544392052, 3.83911766332333044500555981346, 4.83641022168333935087083684979, 5.98550057710762607499762910402, 6.34465995765746847524540248334, 6.81564081461391267496101739549, 7.963939776832785148158508947579, 8.594380094464703376093484678924