| L(s) = 1 | + 2.82i·5-s + 1.73i·7-s + 4.89·11-s + 13-s − 2.82i·17-s + 5.19i·19-s + 4.89·23-s − 3.00·25-s − 5.65i·29-s + 3.46i·31-s − 4.89·35-s + 37-s + 5.65i·41-s + 3.46i·43-s − 4.89·47-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s + 0.654i·7-s + 1.47·11-s + 0.277·13-s − 0.685i·17-s + 1.19i·19-s + 1.02·23-s − 0.600·25-s − 1.05i·29-s + 0.622i·31-s − 0.828·35-s + 0.164·37-s + 0.883i·41-s + 0.528i·43-s − 0.714·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854705209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854705209\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−9.531345510963549863083126791032, −8.819444625732403951224664829238, −7.893339868737661312967550000737, −6.96080511207827257690301636282, −6.41317143934561923967195591340, −5.68931335684947911323347140921, −4.45147334543587499772078685744, −3.45274099738324577098361050550, −2.71807492369847361217513547802, −1.44622892923630363541304710342,
0.789782212133105080676238111218, 1.63517492241153718724265220868, 3.30478726462640934107527356211, 4.25742208313532796767268826147, 4.82231514509249154918294601857, 5.88042799431600256309065630045, 6.78223786931276240965669413526, 7.48427055886027262673591317967, 8.722894785048230225576398480519, 8.895285527451951755009018166687
