L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 3·13-s + 16-s + 3·17-s − 18-s + 2·19-s + 3·23-s − 24-s − 3·26-s + 27-s + 7·29-s − 31-s − 32-s − 3·34-s + 36-s + 4·37-s − 2·38-s + 3·39-s + 5·41-s − 3·43-s − 3·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.832·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.625·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 1.29·29-s − 0.179·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.657·37-s − 0.324·38-s + 0.480·39-s + 0.780·41-s − 0.457·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.136558085\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.136558085\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 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
−7.77909327836352500299774911308, −7.59872898924479954069115411356, −6.55968805080063558805225799482, −6.05257649328496999954777952055, −5.10202905571881481355397918084, −4.22824987861372747254183750412, −3.27862185521101989220592144376, −2.76469986240008805227884851071, −1.61720231748495350569304843286, −0.853104750646525128048413960087,
0.853104750646525128048413960087, 1.61720231748495350569304843286, 2.76469986240008805227884851071, 3.27862185521101989220592144376, 4.22824987861372747254183750412, 5.10202905571881481355397918084, 6.05257649328496999954777952055, 6.55968805080063558805225799482, 7.59872898924479954069115411356, 7.77909327836352500299774911308