L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3.04·7-s − 8-s + 9-s + 10-s + 6.24·11-s − 12-s + 3.04·14-s + 15-s + 16-s + 2.69·17-s − 18-s + 5.82·19-s − 20-s + 3.04·21-s − 6.24·22-s − 5.62·23-s + 24-s + 25-s − 27-s − 3.04·28-s − 5.14·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.15·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.88·11-s − 0.288·12-s + 0.814·14-s + 0.258·15-s + 0.250·16-s + 0.652·17-s − 0.235·18-s + 1.33·19-s − 0.223·20-s + 0.665·21-s − 1.33·22-s − 1.17·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.576·28-s − 0.955·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9406230036\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9406230036\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.91T + 61T^{2} \) |
| 67 | \( 1 - 0.198T + 67T^{2} \) |
| 71 | \( 1 - 0.374T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 + 9.50T + 89T^{2} \) |
| 97 | \( 1 - 0.335T + 97T^{2} \) |
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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
−8.254310396839938562179579564069, −7.31075237809650076807888570509, −6.98665011183212563956694703331, −6.01678383167122941514830628513, −5.78257793118141653121458651401, −4.32964040596805463641252783431, −3.72314610947989989444636360494, −2.91737397049297718024868799860, −1.51440730829124522439258797641, −0.64740718895793163087470704205,
0.64740718895793163087470704205, 1.51440730829124522439258797641, 2.91737397049297718024868799860, 3.72314610947989989444636360494, 4.32964040596805463641252783431, 5.78257793118141653121458651401, 6.01678383167122941514830628513, 6.98665011183212563956694703331, 7.31075237809650076807888570509, 8.254310396839938562179579564069