L(s) = 1 | + (0.951 + 0.309i)2-s + (0.183 − 0.253i)3-s + (0.809 + 0.587i)4-s + (−0.891 + 0.453i)5-s + (0.253 − 0.183i)6-s − i·7-s + (0.587 + 0.809i)8-s + (0.278 + 0.857i)9-s + (−0.987 + 0.156i)10-s + (0.297 − 0.0966i)12-s + (1.87 − 0.610i)13-s + (0.309 − 0.951i)14-s + (−0.0489 + 0.309i)15-s + (0.309 + 0.951i)16-s + 0.902i·18-s + (−1.14 + 0.831i)19-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)2-s + (0.183 − 0.253i)3-s + (0.809 + 0.587i)4-s + (−0.891 + 0.453i)5-s + (0.253 − 0.183i)6-s − i·7-s + (0.587 + 0.809i)8-s + (0.278 + 0.857i)9-s + (−0.987 + 0.156i)10-s + (0.297 − 0.0966i)12-s + (1.87 − 0.610i)13-s + (0.309 − 0.951i)14-s + (−0.0489 + 0.309i)15-s + (0.309 + 0.951i)16-s + 0.902i·18-s + (−1.14 + 0.831i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.890179394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.890179394\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (0.891 - 0.453i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.183 + 0.253i)T + (-0.309 - 0.951i)T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.87 + 0.610i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.280 - 0.863i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (1.04 + 1.44i)T + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 + 0.951i)T^{2} \) |
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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
−10.37622646929521684315276211271, −8.393198244942516690358709597942, −8.131056345861204868583311737431, −7.36982483989745460662721807094, −6.54161797383864509301113536233, −5.83757631590870573473678956610, −4.45849986981699923236418040231, −3.95480635087920498343441836677, −3.13005740740165338049053689735, −1.72390749674610302089145511011,
1.45264672726051789130269908628, 2.82821257875307818588889432457, 3.94638273737825084054930926745, 4.20362938791200721666884555035, 5.47929252920317047447232346336, 6.26219518719346946245889474827, 6.95992286148179343730564677011, 8.319191556700376378328819773381, 8.804721286273900514184579052272, 9.654660614105021222094412605232