L(s) = 1 | − 1.41·2-s + 1.00·4-s + (−0.707 − 0.707i)5-s + 7-s + (1.00 + 1.00i)10-s + (0.707 + 0.707i)11-s + 13-s − 1.41·14-s − 0.999·16-s + (0.707 + 0.707i)17-s + (−1 + i)19-s + (−0.707 − 0.707i)20-s + (−1.00 − 1.00i)22-s + 1.00i·25-s − 1.41·26-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.00·4-s + (−0.707 − 0.707i)5-s + 7-s + (1.00 + 1.00i)10-s + (0.707 + 0.707i)11-s + 13-s − 1.41·14-s − 0.999·16-s + (0.707 + 0.707i)17-s + (−1 + i)19-s + (−0.707 − 0.707i)20-s + (−1.00 − 1.00i)22-s + 1.00i·25-s − 1.41·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5784716532\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5784716532\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−9.405720790434907942929567760904, −8.556930551110096147404930128433, −8.161614390537045278900253634174, −7.63488712013468559705617189396, −6.62584457141504154067158440796, −5.55838622453264946999996681285, −4.38073870248856346466314826275, −3.85317508636110359006920424681, −1.87273442627874359239007591741, −1.22289624250756854025761807587,
0.886300269604360441991794777864, 2.13708036896424846212987182625, 3.45334542279553442180053000133, 4.36489223054557317960488353479, 5.54858590982354557428096285810, 6.79458235235264015242919902032, 7.18008772138738325752682319834, 8.173737243570453025354746091191, 8.612262909857310924564339110100, 9.182693989663691525896000805968