| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 18-s − 19-s + 20-s − 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + 29-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 18-s − 19-s + 20-s − 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.057845273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.057845273\) |
| \(L(1)\) |
\(\approx\) |
\(1.891118950\) |
| \(L(1)\) |
\(\approx\) |
\(1.891118950\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| 97 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.881087712593615372348791328636, −19.87540411009471004248458742799, −18.76637960855691357973020383489, −18.02564581903737052398634606318, −17.20797214392212696321329777824, −16.86819822797810638270615108385, −15.87500773406255154248051088919, −15.01741008119632907782564768230, −14.46172573593572293470023761319, −13.540057904128562299875737904577, −12.79215322085368953303199012008, −12.34632909061107823452616874028, −11.308958202349875533829390640083, −10.665922306443334509310795339278, −10.25150263972958641195074169308, −9.03697853751392250840577323801, −7.682628937728017083638385543497, −7.14303745644277691730756152308, −6.08898410563363402163609614712, −5.54906050397127472692108389263, −4.80315197644414282492808945003, −4.36721233826527012271580285453, −2.71927286955236157412507367140, −2.11508822236903876082764270367, −1.08050231074520278339965730781,
1.08050231074520278339965730781, 2.11508822236903876082764270367, 2.71927286955236157412507367140, 4.36721233826527012271580285453, 4.80315197644414282492808945003, 5.54906050397127472692108389263, 6.08898410563363402163609614712, 7.14303745644277691730756152308, 7.682628937728017083638385543497, 9.03697853751392250840577323801, 10.25150263972958641195074169308, 10.665922306443334509310795339278, 11.308958202349875533829390640083, 12.34632909061107823452616874028, 12.79215322085368953303199012008, 13.540057904128562299875737904577, 14.46172573593572293470023761319, 15.01741008119632907782564768230, 15.87500773406255154248051088919, 16.86819822797810638270615108385, 17.20797214392212696321329777824, 18.02564581903737052398634606318, 18.76637960855691357973020383489, 19.87540411009471004248458742799, 20.881087712593615372348791328636
