| L(s) = 1 | − 4·7-s − 13-s + 2·17-s + 2·29-s − 4·31-s − 6·37-s + 6·41-s − 4·43-s − 4·47-s + 9·49-s − 10·53-s − 2·61-s − 8·67-s − 4·71-s + 6·73-s − 8·79-s + 8·83-s + 6·89-s + 4·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s + 0.485·17-s + 0.371·29-s − 0.718·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s − 1.37·53-s − 0.256·61-s − 0.977·67-s − 0.474·71-s + 0.702·73-s − 0.900·79-s + 0.878·83-s + 0.635·89-s + 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9809588578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9809588578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.60020002725716, −14.88819423717055, −14.41415537189885, −13.77761827834441, −13.24382653977451, −12.73024577138430, −12.31942840558233, −11.78647635080342, −11.01679717376865, −10.43925835741689, −9.920204788416625, −9.435584993132088, −8.955545040071272, −8.228083548579848, −7.502656619362669, −7.014234431551220, −6.333809156668963, −5.933340488967630, −5.149139624839813, −4.476473442962714, −3.537366552258816, −3.242308851896991, −2.441583653670029, −1.504055657780285, −0.3941484884754634,
0.3941484884754634, 1.504055657780285, 2.441583653670029, 3.242308851896991, 3.537366552258816, 4.476473442962714, 5.149139624839813, 5.933340488967630, 6.333809156668963, 7.014234431551220, 7.502656619362669, 8.228083548579848, 8.955545040071272, 9.435584993132088, 9.920204788416625, 10.43925835741689, 11.01679717376865, 11.78647635080342, 12.31942840558233, 12.73024577138430, 13.24382653977451, 13.77761827834441, 14.41415537189885, 14.88819423717055, 15.60020002725716
