L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 2·13-s + 2·17-s + 8·19-s + 21-s + 27-s − 6·29-s + 8·31-s − 33-s − 10·37-s − 2·39-s − 2·41-s − 4·43-s + 4·47-s + 49-s + 2·51-s + 6·53-s + 8·57-s − 4·59-s − 10·61-s + 63-s + 8·71-s − 6·73-s − 77-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 1.83·19-s + 0.218·21-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s − 1.64·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 1.05·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s + 0.949·71-s − 0.702·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.170372453\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.170372453\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.56101044270335, −14.06363598288697, −13.58364374108046, −13.36442229827909, −12.39639295448037, −12.04988335633986, −11.71829442284084, −10.86824677748979, −10.44879964347887, −9.752929374568871, −9.503369456784106, −8.817339963315023, −8.198816285597913, −7.752979090500791, −7.244595266676200, −6.796303953252964, −5.849163466464593, −5.321733825824997, −4.867318696745898, −4.135228266002699, −3.313952153390839, −3.019946437181805, −2.122043145114761, −1.487202299010265, −0.6218255551986557,
0.6218255551986557, 1.487202299010265, 2.122043145114761, 3.019946437181805, 3.313952153390839, 4.135228266002699, 4.867318696745898, 5.321733825824997, 5.849163466464593, 6.796303953252964, 7.244595266676200, 7.752979090500791, 8.198816285597913, 8.817339963315023, 9.503369456784106, 9.752929374568871, 10.44879964347887, 10.86824677748979, 11.71829442284084, 12.04988335633986, 12.39639295448037, 13.36442229827909, 13.58364374108046, 14.06363598288697, 14.56101044270335