| L(s) = 1 | − 1.31i·2-s + i·3-s + 0.274·4-s + 1.31·6-s − 4.19i·7-s − 2.98i·8-s − 9-s − 0.167·11-s + 0.274i·12-s − 3.39i·13-s − 5.50·14-s − 3.37·16-s − 4.57i·17-s + 1.31i·18-s − 3.78·19-s + ⋯ |
| L(s) = 1 | − 0.928i·2-s + 0.577i·3-s + 0.137·4-s + 0.536·6-s − 1.58i·7-s − 1.05i·8-s − 0.333·9-s − 0.0505·11-s + 0.0792i·12-s − 0.941i·13-s − 1.47·14-s − 0.843·16-s − 1.10i·17-s + 0.309i·18-s − 0.867·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301105001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301105001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.31iT - 2T^{2} \) |
| 7 | \( 1 + 4.19iT - 7T^{2} \) |
| 11 | \( 1 + 0.167T + 11T^{2} \) |
| 13 | \( 1 + 3.39iT - 13T^{2} \) |
| 17 | \( 1 + 4.57iT - 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 - 8.31iT - 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 + 2.07iT - 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 11.6iT - 43T^{2} \) |
| 47 | \( 1 - 9.99iT - 47T^{2} \) |
| 53 | \( 1 + 1.07iT - 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 + 12.2iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8.67iT - 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 4.06T + 89T^{2} \) |
| 97 | \( 1 - 2.78iT - 97T^{2} \) |
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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
−9.402260159721736674419549692574, −7.87554962446109160138049651577, −7.41641577422392559786553297381, −6.52859040914509570752577230064, −5.42084061538923502587795597458, −4.41325217949033435161845788543, −3.62681529225100231387171968528, −3.01763911288836710505981570200, −1.64618669841945152377056796999, −0.43534788664112353404778212823,
2.04801641609444402547872555129, 2.32432425011133566265420616525, 3.91341887783670172840213963108, 5.22493054448314905456854905693, 5.75975484584394076768171149911, 6.60084937206758214524313767227, 6.95021941553433885478255571544, 8.237096148679893669576941196656, 8.527380194109442375503615751222, 9.148359165950020678513877212538
