| L(s) = 1 | + 2.41i·2-s + 0.414i·3-s − 3.82·4-s − 0.999·6-s − 4.41i·8-s + 2.82·9-s − 0.828·11-s − 1.58i·12-s + 4.82i·13-s + 2.99·16-s + 4.82i·17-s + 6.82i·18-s − 2.82·19-s − 1.99i·22-s − 0.414i·23-s + 1.82·24-s + ⋯ |
| L(s) = 1 | + 1.70i·2-s + 0.239i·3-s − 1.91·4-s − 0.408·6-s − 1.56i·8-s + 0.942·9-s − 0.249·11-s − 0.457i·12-s + 1.33i·13-s + 0.749·16-s + 1.17i·17-s + 1.60i·18-s − 0.648·19-s − 0.426i·22-s − 0.0863i·23-s + 0.373·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044972840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044972840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 3 | \( 1 - 0.414iT - 3T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 0.414iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 3.58iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 - 9.58iT - 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 - 0.828iT - 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−10.00935751800567235657828884353, −9.114941465279512578307798663928, −8.556923705454730968340843437011, −7.64880539479695985312465679328, −6.89376350252349983677760712199, −6.34516014478940266369847708000, −5.36211021035403937791716824953, −4.44892805276090377480804048687, −3.85035116590754950418824587376, −1.83405806617274399516706734816,
0.45040224235375158783925959240, 1.67289275833556253685634577890, 2.72828884805485961090062048337, 3.58120989908465462217839182752, 4.60830421594009200086644990036, 5.39828433830828723313812992531, 6.78261874460367081745409423622, 7.71476431445505379837192111067, 8.614946570303044132203352376306, 9.566733530969615334007087042243
