| L(s) = 1 | − 5i·3-s − 19i·7-s + 2·9-s − 11·11-s + 62i·13-s + 19i·17-s + 131·19-s − 95·21-s − 138i·23-s − 145i·27-s + 79·29-s + 217·31-s + 55i·33-s − 91i·37-s + 310·39-s + ⋯ |
| L(s) = 1 | − 0.962i·3-s − 1.02i·7-s + 0.0740·9-s − 0.301·11-s + 1.32i·13-s + 0.271i·17-s + 1.58·19-s − 0.987·21-s − 1.25i·23-s − 1.03i·27-s + 0.505·29-s + 1.25·31-s + 0.290i·33-s − 0.404i·37-s + 1.27·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.133915258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133915258\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 5iT - 27T^{2} \) |
| 7 | \( 1 + 19iT - 343T^{2} \) |
| 13 | \( 1 - 62iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 19iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 131T + 6.85e3T^{2} \) |
| 23 | \( 1 + 138iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 79T + 2.43e4T^{2} \) |
| 31 | \( 1 - 217T + 2.97e4T^{2} \) |
| 37 | \( 1 + 91iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 158T + 6.89e4T^{2} \) |
| 43 | \( 1 + 120iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 546iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 439iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 290T + 2.05e5T^{2} \) |
| 61 | \( 1 + 373T + 2.26e5T^{2} \) |
| 67 | \( 1 - 728iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 709T + 3.57e5T^{2} \) |
| 73 | \( 1 + 850iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 58iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 753T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3iT - 9.12e5T^{2} \) |
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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
−9.182874397614046213219672756627, −8.184571189265140688547422724105, −7.37441516047617110819443821108, −6.89294266129683964458650492023, −6.08927058344064665508737410560, −4.74617831694274079325337804945, −3.99143238173518325980557068413, −2.64685930907670850425351787366, −1.48520898346069814337900320792, −0.60687495269449458939911215333,
1.12954629371681478005902615964, 2.77490149698982191629382029703, 3.38865830170824916348668047155, 4.73576621307274524605533093751, 5.32366442241171258207239678966, 6.09980670241380036493376796773, 7.46688280919689240450089947947, 8.101396416966989438976490395673, 9.197253607833276238337273955519, 9.694782329938680211492358975341
