L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + (−0.5 + 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (−0.707 + 0.707i)18-s + (−0.965 + 0.258i)22-s + (−1.22 − 1.22i)23-s + 1.73·29-s + (−1.22 + 1.22i)37-s + (0.707 − 0.707i)43-s − 1.73i·46-s + 1.00i·49-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + (−0.5 + 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (−0.707 + 0.707i)18-s + (−0.965 + 0.258i)22-s + (−1.22 − 1.22i)23-s + 1.73·29-s + (−1.22 + 1.22i)37-s + (0.707 − 0.707i)43-s − 1.73i·46-s + 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.801632878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.801632878\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 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
−9.602425835678759839376886930620, −8.347227561603601960353654643875, −8.015168285218385262371529013193, −7.03058185419447384143089564355, −6.30424903774231907587394106573, −5.34756074085885972361469411069, −4.86115429932486672353366002324, −4.22162507074858748592737711805, −2.62412005679996567531933697609, −1.73202468792968190109525670572,
1.21639681428285370711287025162, 2.49334557247972217385904514221, 3.54730265318532926737536231250, 4.02201433090993204038494427099, 5.03489902729095726799909507247, 5.82445932643668838864506844204, 6.88277911002229441516377273589, 7.82629769864416755633766877862, 8.313775977605543410275583287431, 9.293651425201726605739648226486