| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s + 13-s − 15-s + 16-s + 6·17-s − 18-s − 20-s + 4·23-s − 24-s + 25-s − 26-s + 27-s + 10·29-s + 30-s − 32-s − 6·34-s + 36-s + 6·37-s + 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.223·20-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.85·29-s + 0.182·30-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.584842295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.584842295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.47208265816272, −12.05327197691520, −11.50422595793551, −11.09912731114773, −10.63199036301027, −10.03933050877802, −9.885175035649576, −9.222044072063888, −8.865538668189109, −8.338558980772501, −7.979503730188513, −7.581798352951796, −7.188170919158915, −6.443919113679314, −6.292185596598760, −5.476369159959533, −4.993468870211047, −4.439969417189079, −3.814241419428725, −3.286437736055302, −2.884105389435833, −2.365149949686859, −1.605148898811573, −0.8945347748961787, −0.6938642701723337,
0.6938642701723337, 0.8945347748961787, 1.605148898811573, 2.365149949686859, 2.884105389435833, 3.286437736055302, 3.814241419428725, 4.439969417189079, 4.993468870211047, 5.476369159959533, 6.292185596598760, 6.443919113679314, 7.188170919158915, 7.581798352951796, 7.979503730188513, 8.338558980772501, 8.865538668189109, 9.222044072063888, 9.885175035649576, 10.03933050877802, 10.63199036301027, 11.09912731114773, 11.50422595793551, 12.05327197691520, 12.47208265816272
