| L(s) = 1 | + (−0.707 + 0.707i)5-s + (1 + i)13-s + (1.41 + 1.41i)17-s − 1.00i·25-s − 1.41·29-s + (−1 + i)37-s − 1.41i·41-s − i·49-s + (1.41 − 1.41i)53-s − 1.41·65-s + (−1 − i)73-s − 2.00·85-s + 1.41·89-s + (−1 + i)97-s − 1.41i·101-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)5-s + (1 + i)13-s + (1.41 + 1.41i)17-s − 1.00i·25-s − 1.41·29-s + (−1 + i)37-s − 1.41i·41-s − i·49-s + (1.41 − 1.41i)53-s − 1.41·65-s + (−1 − i)73-s − 2.00·85-s + 1.41·89-s + (−1 + i)97-s − 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8744477239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8744477239\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−10.69586414033327449488954340451, −10.09526613516142092490867168529, −8.875895662919085980133728579828, −8.158999394857400877990101158500, −7.24121539184512390265557339016, −6.41368655815196284656496659776, −5.46105895518190051972033628609, −3.94681818133471158596379612285, −3.48464899534862767580549396055, −1.79867177698340179480444143221,
1.09226915015899176819216067128, 3.04288437003052259917622483598, 3.95107740013189464419388771609, 5.17495173136970089075037723889, 5.82121470846177135671677441346, 7.31779022604041152949830528562, 7.83549547061645042121765394623, 8.785891869991726383636566755147, 9.534181708363558749196133230669, 10.56789003391955680103010733342
