L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + 5-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)10-s + 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 + i)17-s − 1.41i·19-s − 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (0.707 − 0.707i)28-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + 5-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)10-s + 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 + i)17-s − 1.41i·19-s − 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (0.707 − 0.707i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9291878800\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9291878800\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + 2iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.832926153162524570280014628211, −9.075130608323551653660838450450, −8.693369829584176818558533126611, −7.54448674631309993284748732415, −6.82805233357577585111418683528, −6.01683665928884551687878272564, −5.13729643570304754357811602424, −4.49499505462255821884902086408, −2.33040060687394007776341711173, −1.75586070676143837343040325415,
1.12183159499626468381880677725, 2.22855267312832248037897920716, 3.34491376570338424905121438394, 4.38529663055047805334252454283, 5.52083872024425997673813104689, 6.49837795726161038754314142375, 7.49560741251107824503204334781, 8.263193592620122600055619196075, 9.033484104036436828095170000236, 9.722403030498742022998025751770