| L(s) = 1 | + (−1 − i)2-s + 2i·4-s − 1.73i·5-s + (2 − 2i)8-s + (−1.73 + 1.73i)10-s − i·11-s + 3.46i·13-s − 4·16-s + 1.73i·17-s + 5.19·19-s + 3.46·20-s + (−1 + i)22-s + i·23-s + 2.00·25-s + (3.46 − 3.46i)26-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s − 0.774i·5-s + (0.707 − 0.707i)8-s + (−0.547 + 0.547i)10-s − 0.301i·11-s + 0.960i·13-s − 16-s + 0.420i·17-s + 1.19·19-s + 0.774·20-s + (−0.213 + 0.213i)22-s + 0.208i·23-s + 0.400·25-s + (0.679 − 0.679i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.189606397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189606397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 - 5.19iT - 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 14iT - 71T^{2} \) |
| 73 | \( 1 - 8.66iT - 73T^{2} \) |
| 79 | \( 1 + 9iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 15.5iT - 89T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
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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
−9.185304582467199686391351913459, −8.666071664420291904168486894168, −7.78576811091325290067089999008, −7.09103375691693862996366538239, −5.99528243713056950803383651794, −4.88980461809374284139112356957, −4.06146484947744979374346291766, −3.10004910286515755511790684133, −1.87817368842723006370202751786, −0.872182168283802581494826426066,
0.835315931188436297924140425206, 2.33095361418293132161863258608, 3.35769054750328220706337392777, 4.74156242101763423484470598179, 5.56114543391741238765336201377, 6.32228063502552642530878523169, 7.29476750798436871974007871446, 7.57987629864807306330883108038, 8.569997210141214568932619407132, 9.415031666698429248866399554469
