L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + i·7-s + 1.00i·9-s + 1.41·11-s − 1.00·15-s + (0.707 − 0.707i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41i·29-s + 2i·31-s + (−1.00 − 1.00i)33-s + (0.707 + 0.707i)35-s + (0.707 + 0.707i)45-s − 49-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + i·7-s + 1.00i·9-s + 1.41·11-s − 1.00·15-s + (0.707 − 0.707i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41i·29-s + 2i·31-s + (−1.00 − 1.00i)33-s + (0.707 + 0.707i)35-s + (0.707 + 0.707i)45-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.232452340\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.232452340\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 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
−8.739693109551092308687939849226, −8.237518684135137311410493135255, −6.88534397447320625573183634673, −6.64680864283811820581289763685, −5.64060467304776256336655797492, −5.27409178840062880015855129755, −4.36764642988661584798371395177, −3.04580474335413517861486799073, −1.87219009593183543242796165403, −1.24973401361455021484305002856,
0.973027253471665838444414465238, 2.30594605071210871582426499340, 3.68271564241593316547300509276, 4.00508583512675145322602768476, 4.99406100000956186268089200398, 6.04811878382530264620116410581, 6.38663482944322760886361868085, 7.12241744041476748099488694228, 7.994566622135988761074239981782, 9.258903019933022436290966426891