| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 0.381i·7-s + i·8-s − 9-s + 1.38·11-s + i·12-s + 2.47i·13-s − 0.381·14-s + 16-s + 3.23i·17-s + i·18-s − 7.70·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.144i·7-s + 0.353i·8-s − 0.333·9-s + 0.416·11-s + 0.288i·12-s + 0.685i·13-s − 0.102·14-s + 0.250·16-s + 0.784i·17-s + 0.235i·18-s − 1.76·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216278722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216278722\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.381iT - 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 - 2.47iT - 13T^{2} \) |
| 17 | \( 1 - 3.23iT - 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 5.70iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 9.09iT - 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 12.4iT - 73T^{2} \) |
| 79 | \( 1 - 3.38T + 79T^{2} \) |
| 83 | \( 1 - 8.85iT - 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 5.61iT - 97T^{2} \) |
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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
−8.513686646735142690944806102630, −8.022890954340519638780663219421, −6.85967288333667505161520049687, −6.43619333283203675267280399691, −5.56167598858894034674322289281, −4.32018699830566496315850034025, −4.05212000828020489745152026707, −2.71779136801374832016358090500, −2.02143883199781535915446479876, −1.01253104172303924867648600466,
0.41129575226261045029737384163, 2.03590375370981372238896888714, 3.24250830238277051580829167914, 3.97302534598986944478945281617, 4.91596546278039138270145106310, 5.39931963248966999601693959755, 6.40300272300965170759674893777, 6.86202309122341495256972294675, 7.900094904686301360764218554377, 8.518665926117470303618857751210
