| L(s) = 1 | − 3.12i·3-s + 1.51i·7-s − 6.76·9-s + 4.24·11-s + 4.15i·13-s + 3.51i·17-s + 19-s + 4.73·21-s − 8.73i·23-s + 11.7i·27-s − 1.45·29-s − 4.96·31-s − 13.2i·33-s − 7.60i·37-s + 12.9·39-s + ⋯ |
| L(s) = 1 | − 1.80i·3-s + 0.572i·7-s − 2.25·9-s + 1.28·11-s + 1.15i·13-s + 0.852i·17-s + 0.229·19-s + 1.03·21-s − 1.82i·23-s + 2.26i·27-s − 0.270·29-s − 0.892·31-s − 2.31i·33-s − 1.25i·37-s + 2.07·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.459487798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.459487798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.12iT - 3T^{2} \) |
| 7 | \( 1 - 1.51iT - 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.15iT - 13T^{2} \) |
| 17 | \( 1 - 3.51iT - 17T^{2} \) |
| 23 | \( 1 + 8.73iT - 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 7.60iT - 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 8.31iT - 43T^{2} \) |
| 47 | \( 1 + 5.28iT - 47T^{2} \) |
| 53 | \( 1 - 0.155iT - 53T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 + 4.49T + 61T^{2} \) |
| 67 | \( 1 + 7.43iT - 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 0.310T + 79T^{2} \) |
| 83 | \( 1 + 8.96iT - 83T^{2} \) |
| 89 | \( 1 + 0.719T + 89T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
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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
−8.209311645943471055362800098633, −7.21255618833795909528357174289, −6.77435163943699376179243606848, −6.21390545134538307425292591799, −5.52493737294188245204323589832, −4.30000658686166837764257869129, −3.34144194769584065643569725860, −1.99705044600949965226312078625, −1.84532897939226261473956125606, −0.45307212494283683265168155256,
1.17584901098911734538775764151, 2.90191346162540345111680790215, 3.57585510684838047568963495053, 4.08609229368864350604986600749, 5.08517290111110321924192567687, 5.47219334891452742009069214968, 6.48563097191031575804088776563, 7.43674910045826543919413625320, 8.233767660586758626627529036797, 9.103639385913340344282876946568
