L(s) = 1 | + i·2-s − 2.44i·3-s − 4-s + 2.44·6-s − 0.449i·7-s − i·8-s − 2.99·9-s + 2·11-s + 2.44i·12-s − 2.44i·13-s + 0.449·14-s + 16-s − 2i·17-s − 2.99i·18-s − 1.55·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.41i·3-s − 0.5·4-s + 0.999·6-s − 0.169i·7-s − 0.353i·8-s − 0.999·9-s + 0.603·11-s + 0.707i·12-s − 0.679i·13-s + 0.120·14-s + 0.250·16-s − 0.485i·17-s − 0.707i·18-s − 0.355·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.211932376\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.211932376\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 7 | \( 1 + 0.449iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 + 2.44iT - 23T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 1.44iT - 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 0.898iT - 43T^{2} \) |
| 47 | \( 1 + 3.89iT - 47T^{2} \) |
| 53 | \( 1 + 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + 8.55iT - 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 - 1.34iT - 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 8.44T + 89T^{2} \) |
| 97 | \( 1 + 7.34iT - 97T^{2} \) |
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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
−8.931133470799966285867921755611, −8.242251604341794977444603964977, −7.51096503197804036898234892367, −6.87463634680763686940395310552, −6.25297418006538971067670274510, −5.38659916503155576644925348815, −4.27614888635459448993043043867, −3.02016948314528278590651837676, −1.72554745935703660930138230652, −0.49700477550509189015436159945,
1.62013578213626474716980281381, 2.96375911662350427182263275907, 3.92776772645798855927164009089, 4.40907723764914993691765018579, 5.37360136234552349302611584651, 6.29222070660684011500727322197, 7.49679986597995208862348698193, 8.720333017252885705808512429768, 9.119575156057513304525276921743, 9.824782191007785247335856860283