| L(s) = 1 | − 2.59i·2-s + i·3-s − 4.74·4-s + 2.59·6-s − 3.28i·7-s + 7.12i·8-s − 9-s + 4.30·11-s − 4.74i·12-s − 3.46i·13-s − 8.52·14-s + 9.02·16-s + 5.44i·17-s + 2.59i·18-s + 7.63·19-s + ⋯ |
| L(s) = 1 | − 1.83i·2-s + 0.577i·3-s − 2.37·4-s + 1.06·6-s − 1.24i·7-s + 2.52i·8-s − 0.333·9-s + 1.29·11-s − 1.36i·12-s − 0.959i·13-s − 2.27·14-s + 2.25·16-s + 1.32i·17-s + 0.612i·18-s + 1.75·19-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
| \(L(1)\) |
\(\approx\) |
\(1.484574057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484574057\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.59iT - 2T^{2} \) |
| 7 | \( 1 + 3.28iT - 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.44iT - 17T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 23 | \( 1 + 5.04iT - 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 - 1.89iT - 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 + 3.20iT - 43T^{2} \) |
| 47 | \( 1 + 6.28iT - 47T^{2} \) |
| 53 | \( 1 + 2.51iT - 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 + 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 + 1.64iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 5.01iT - 83T^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 + 2.57iT - 97T^{2} \) |
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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
−9.189160122517000221355620249491, −8.450978990161785733018233661012, −7.52026394347120932685946308102, −6.27568096806360778111340450533, −5.12276875517961248681601750597, −4.25681413431806615545635935698, −3.68130539689149192698386422338, −3.02705988199561825107811055493, −1.55763173465553852705134698076, −0.66405048079214833533605013133,
1.27505966423836634017207256362, 2.94090302642232033936320790251, 4.21860508148772062697153251528, 5.19687803847586207606229057501, 5.79256689041627218948565538203, 6.51844473345884977764943526082, 7.23789569590591979470834642300, 7.73281244565067630899982933385, 8.856515033138050164242085978164, 9.219325154731166156938221963999
