| L(s) = 1 | + 0.209·2-s − 0.956·4-s − 1.82·5-s − 0.408·8-s − 0.381·10-s − 1.33·11-s + 0.870·16-s + 17-s + 1.74·20-s − 0.279·22-s + 2.33·25-s + 1.95·29-s − 0.209·31-s + 0.591·32-s + 0.209·34-s + 0.747·40-s + 1.27·44-s + 49-s + 0.488·50-s + 2.44·55-s + 0.408·58-s − 61-s − 0.0437·62-s − 0.747·64-s − 67-s − 0.956·68-s − 0.618·71-s + ⋯ |
| L(s) = 1 | + 0.209·2-s − 0.956·4-s − 1.82·5-s − 0.408·8-s − 0.381·10-s − 1.33·11-s + 0.870·16-s + 17-s + 1.74·20-s − 0.279·22-s + 2.33·25-s + 1.95·29-s − 0.209·31-s + 0.591·32-s + 0.209·34-s + 0.747·40-s + 1.27·44-s + 49-s + 0.488·50-s + 2.44·55-s + 0.408·58-s − 61-s − 0.0437·62-s − 0.747·64-s − 67-s − 0.956·68-s − 0.618·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5837303306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5837303306\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 0.209T + T^{2} \) |
| 5 | \( 1 + 1.82T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.33T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.95T + T^{2} \) |
| 31 | \( 1 + 0.209T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.95T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.037976622326067240499765743371, −8.331934732710441045542219608331, −7.85023194350005601359916680656, −7.24886909872050375363704445482, −5.97209407022927340533232441283, −4.95889361188729678718255792119, −4.50727833265194909553852491258, −3.53157841189313142122409605515, −2.90696716456792470049807070378, −0.71112688066286858328326085141,
0.71112688066286858328326085141, 2.90696716456792470049807070378, 3.53157841189313142122409605515, 4.50727833265194909553852491258, 4.95889361188729678718255792119, 5.97209407022927340533232441283, 7.24886909872050375363704445482, 7.85023194350005601359916680656, 8.331934732710441045542219608331, 9.037976622326067240499765743371
