L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.382 − 0.923i)5-s + (0.707 − 0.707i)8-s + i·9-s + (−0.923 + 0.382i)10-s + (0.541 + 0.541i)13-s − 1.00·16-s + (1.30 − 1.30i)17-s + (0.707 − 0.707i)18-s + (0.923 + 0.382i)20-s + (−0.707 − 0.707i)25-s − 0.765i·26-s + 1.41i·29-s + (0.707 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.382 − 0.923i)5-s + (0.707 − 0.707i)8-s + i·9-s + (−0.923 + 0.382i)10-s + (0.541 + 0.541i)13-s − 1.00·16-s + (1.30 − 1.30i)17-s + (0.707 − 0.707i)18-s + (0.923 + 0.382i)20-s + (−0.707 − 0.707i)25-s − 0.765i·26-s + 1.41i·29-s + (0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8004588150\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8004588150\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.382 + 0.923i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 17 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 1.84iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.765iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 0.765T + T^{2} \) |
| 97 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
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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
−10.08765788612835161893903956141, −9.188881973054896513902919860519, −8.675212365688477892806137412145, −7.76153791890103110407775420540, −7.02062173935273611914007705228, −5.53589399508718745090006345703, −4.78766880180330098907302900093, −3.61503778386884927738984171518, −2.34401432727634957626441695258, −1.23749399426000847566269155581,
1.38517032279147332638842511162, 2.95318939988085417080955188054, 4.09531495167764112767148498088, 5.77263621626149004081369100369, 6.02775360728284830219978160400, 6.94451484400766008536393251843, 7.86741368328406857375253943659, 8.533660398460043072581622227013, 9.720135230758724370629318760980, 10.00354727942733301508943784645