| L(s) = 1 | + 5-s + 4.35i·7-s − 4.35i·11-s − 7·17-s − 4.35i·19-s − 8.71i·23-s − 4·25-s + 4.35i·35-s − 13.0i·43-s − 4.35i·47-s − 12.0·49-s − 4.35i·55-s + 15·61-s + 11·73-s + 19.0·77-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.64i·7-s − 1.31i·11-s − 1.69·17-s − 0.999i·19-s − 1.81i·23-s − 0.800·25-s + 0.736i·35-s − 1.99i·43-s − 0.635i·47-s − 1.71·49-s − 0.587i·55-s + 1.92·61-s + 1.28·73-s + 2.16·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.158836061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158836061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 4.35iT - 7T^{2} \) |
| 11 | \( 1 + 4.35iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 23 | \( 1 + 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 13.0iT - 43T^{2} \) |
| 47 | \( 1 + 4.35iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.789203814468410657458081175564, −8.242811409315443638822099337603, −6.83371142556904513206406479624, −6.32631396517094626619036615991, −5.57329956825449175714035398194, −4.91508712412840842965599357449, −3.78582158723682244956314339667, −2.50702635510614936132097004729, −2.25068134317812301064841386519, −0.36173015810441053873242161007,
1.33622192729485105473819296935, 2.15851289610000563168969067959, 3.59742293432366150954159283400, 4.24759561997224327573187415089, 4.95948616662942962246825509142, 6.08535659431749085987035216836, 6.82688651803480569959002336625, 7.48234465963839701103035147954, 8.020252288282131077617668391171, 9.253722077343463584061172115274
