| 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
−8.594380094464703376093484678924, −7.963939776832785148158508947579, −6.81564081461391267496101739549, −6.34465995765746847524540248334, −5.98550057710762607499762910402, −4.83641022168333935087083684979, −3.83911766332333044500555981346, −3.29850739635160258775544392052, −2.02503686888088496975828805662, −1.36391992190738093686017882265,
0.65206479458566418174700533289, 1.60761459007167959116718234904, 2.74528810942945675140741171774, 3.52664668355176439151517433165, 4.79345489475214540829469418806, 5.04207174166276388672326087824, 6.15356022866449451091036729266, 6.75464605942904174771816777538, 7.30564250069470430599335697643, 8.528704115925660888811458962046
